study algebra aspect kontsevich integr gen funct intersect the modul spac review der virasoro constraint intersect numb kontsevich integr main theorem expand charact schur funct proof first part theorem grassman matrix airy equ virasoro highest weight condit gen expand singul behavio painlev equ gen high degr pot
canon structure class linear sigm model rieman symmet spac constitut gen class class linear sigm model known integr shown govern funda poisson bracket rel fit matrix form ultraloc integr model first discuss maillet mat comput explicit field depend satisfy funda poisson bracket rel express term num matrix propos poisson bracket tak togeth repres condit kind algebr class model replac class yang baxt algebr govern canon structure ultraloc model poisson bracket transit mat also comput not regul problem assocy definit poisson bracket monodrom mat discuss
show construct start quas hopf algebr quas quant group inv knot link cas inv giv ris inv three manifold obtain surgery along link hap finit dimend quas quant group whos definit involv finit group cocyc first study dijkgraaf pasquy roch tre exampl detail argu cas inv agr partit funct topolog field the dijkgraaf wit depend dat
recal class irreduc repres giv fus rul repres also consid problem mat intertwin diff ord tens produc repres satisfy altogeth yang baxt equ
lect not giv introduc account approach cohomolog field the atiyah jeffrey bas construct gauss shap thom form matha quil top cov expl matha quil form finit dimend vect bundl definit regul eul numb infinit dimend vect bundl interpret supersymmet quant mech regul eul numb loop spac atiyah jeffrey interpret donaldson the construct topolog gaug the infinit dimend vect bundl spac connect
carry study alexand conway inv quant field the point view start cit first discuss detail mat sup model obtain level integ finit dimend repres modul group remark property matrix el infinit moreov typ atyp repres wel indecompos block mix trunt maxim atyp repres advoc rec pap consist main topolog apply work comput alexand inv manifold link manifold inv manifold seem depend triv level stil contain interest topolog inform seifert manifold inst coincid ord first homolog group exampl inv link manifold giv exhibit interest arithmet property
yang baxt equ addit rel ad reflect equ appear vary plac without spect paramet exampl fact scat half lin integr lat model period bound condit commut diff geometry quant group study form spect paramet independ reflect equ chos requir solv comod respect quant group coact leav inv reflect equ vary known solv yang baxt equ giv const solv reflect equ vary quadr algebra defin reflect equ also giv explicit
approach correl funct pres model ant ferroelect regim method bas rec real quant affin symmetry us vertex op boson repres lat integr formul found correl funct arbit loc op spec cas reproduc spont stag pol obtain ear baxt
extery der in der der introduc quant group found construct mat determin un induc calcul found
obtain clos form av polynom tak hermit mat gauss meas involv kontsevich integr prov conject wit en express analog av ful cub pot meas der partit funct respect trac invers pow extern argu proof bas el algebra id involv inv polynom linear group clos rel gen schur funct
vassiliev gusarov link inv finit typ known clos rel perturb the chern simon the ord clar perturb nat link inv introduc algebr infin contain el satisfy us braid group rel el satisfy epsilon epsilon form vary regard meas fail eq topolog el sign cross show larg class link inv finit typ correspond homog markov trac infin sketch poss apply link inv finit typ manifest diffeomorph inv perturb the quant grav loop repres
giv pedagog introduc diff calcul quant group stressing stag connect class cas rightarrow limit der contract op form tens field found explicit form cart maur equ pres exampl bicov diff calcul quant group giv detail soft quant group consid introduc curv satisfy bianch id bas ingredy construct grav gaug the
quant mat known every matrix obey quant yang baxt equ also known vect giv correspond zamalodchikov algebr develop interpret detail distinct form algebr vect covect ten algebr homomorph quant mat real tens produc quant vect quant covect in produc quant covect quant vect transform scal show endow necess braid stat braid tens produc ten real braid mat introduc prevy in produc lead inv quant trac introduc braid stat lead ful cov quant braid linear algebr braid group obtain conjug imposs quant group obtain
braid group braid mat novel algebra structures liv braid quasitens categ gen sup group sup mat cas braid stat construct braid group vert standard quant group gen matrix braid coproduc delt ten self dual apply deg sklyanin algebr shown isomorph braid mat braid commut bialgebr braid categ second apply show quant doubl also known quant lorentz group semidirect produc algebr cop also semidirect produc coalgebr braid stat find vary result typ doubl gen quant group sem class limit doubl algebra poisson group
introduc larg class bicov diff calcul quant group assocy inv el exampl deform trac el recov woronowicz calcul gen obtain sequ diff calcul quant group bas the correspond braid group regul solv qyb
us hirot method soliton construct affin tod field the bas simply lac affin algebra consid automorph simply lac dynkin diagram solv remain algebra twist wel untw deduc
funct form class gen zero curv integr hierarchy part diff equ construct class includ drinfel sokolov hierarchy direct rel vary zero curv form funct est form also clar connect zero curv hierarchy hirot typ hierarchy wakimoto
restrict solid solid model ant ferromagnet regim study framework quant affin algebra follow lin develop rec vertex model repres theoret pict pres structure spac stat loc op cre annihil op quas partic defin us vertex op commut rel calc
quadr algebra rel reflect equ introduc quant group comod algebra quant group tak exampl property algebra cent repres real real form fus proc wel gen discuss
phys interpret main not quant group the coproduc repres corepres act coact discuss us simplest exampl deform object quant group quant algebr oscil cov algebr appropry reduc cov algebr second rank tens giv ris algebra oscil sphere spec cov algebr rel reflect equ correspond braid group spac nont topolog
show sup chern simon the loop exact provid direct proof rel alexand polynom analys reidem tors process comput explicit tors len spac seifert manifold us surgery mat wess zumino wit model rec determin complet agr known result quant field the alexand polynom provid thu model triv topolog cont idea forward wit jon polynom explicit check finit simpl presum gen aspect compact group lik mod rel chern simon wess zumino wit the also illust com clos rel cas
gen method obtain linear diff equ polynom solv propos method bas equ spect problem el univers envelop algebr algebr project repres possess inv subspac spect problem certain linear diff op vary coefficy shown gen polynom solv part diff equ occ cas superalgebra polynom solv matrix diff equ quant algebra giv ris polynom solv finit diff equ particul known class orthogon polynom appear consid act exampl polynom connect project repres brief discuss
gen drinfeld sokolov hierarchy construct loc reduc hamilton flow gen monodrom inv dual loop algebr follow ear work groot reduc bas upon grad regul el arbit heisenberg subalgebra consid show cas nontw loop algebr grad regul el ex heisenberg subalgebra correspond eith partit eq numb eq numb plu prov reduc belong grad regul el cas yield tim matrix vert gelfand dickey hierarchy gen scal cas consid method util throughout analys form reduc entir within hamilton framework provid class matrix approach lead simpl ev
geometry supermanifold provid structure vect field satisfy structure symplect structure structure volum el vary combin structures study result apply analys batalin vilkovisky approach quant gaug the particul semiclass approxim approach express term reidem tors
paramet quadr deform quant oscil algebr paramet deform heisenberg subalgebr consid infinit dimend fock mod repres pres root un contain nul vect reduc finit dimend repres cyc nilpot unit repres discuss wit deform deform infinit dimend algebra construct heisenberg algebr gen deform centreless virasoro algebr root un ment fin symmetry deform heisenberg algebr explicit construct
propos correl funct inhomog eight vertex model ant ferroelect regim satisfy system diff equ respect spect paramet solv simplest diff equ obtain express spont stag pol conject baxt kelland discuss also rel construct vertex op lat
algebr monodrom mat trigonomet mat study shown gen finit dimend polynom irreduc repres algebr equ tens produc op cocommut repres discuss spec class repres fact repres introduc intertwin cocommut fact repres writ boltzman weight chir pot model consid algebr gen noncommut entry matrix satisfy fam bilinear rel origin quant invers scat method matrix hist reason algebr cal algebr monodrom mat simpl finit dimend algebr inv matrix algebr monodrom mat prop spec giv yang introduc drinfeld correspond trigonomet matrix algebr clos connect zero level ellipt matrix algebr monodrom mat giv ris sklyanin algebr pap shal study algebra monodrom mat trigonomet mat root finit dimend cyc irreduc polynom repres intertwin discuss
four lect giv nanka institut mathem tianjin chin april pres el introduc quant integr model aim mathem phys mathem stress mad algebra aspect the problem determin spectr quant integr mot magnet chain us bas exampl lect devot detail exposit funct beth ansatz techn altern algebra beth ansatz rel sep vary method poss extend discuss
funda problem study the hamilton integr system integr equ mot construct act angl vary third problem howev ad list sep vary though much simpl oth import rel quant integr sep vary construct magnet chain exampl integr model assocy nonhyperellipt algebra curv
prov id valu rog dilogarithm funct describ connect id spectr conform field the
scal level spac distribut funct bulk spectr random matrix model tim hermit mat going limit infty lead fredholm determin sin kernel simil scal limit edg spectr lead airy kernel pap der analog airy kernel follow property sin kernel complet integr system found jimbo miw sato express cas singl interv fredholm determin term painlev transcend ex commut diff op fact op us der asymptot gen prob interv contain prec eigenvalu
reflect equ consist extend yang baxt equ addit el cal reflect matrix matrix exampl describ condit fact scat half lin lik giv condit fact scat entir lin gen defin quadr algebra yang baxt algebra us intend discuss quant group simil defin quadr algebra reflect equ algebra enjoy vary remark property inherit foc vary property particul quant group comod property gen sery solv compos known solv extend cent el
polyakov rec show conform field the describ dimend turb construct infinit hierarchy solv const enstroph flux cascad const energy flux cascad conclud spec concern stabl phys mean solv
prec definit quant integr sep vary serv pract substitut quant integr model gen yang canon coordin conjug op construct satisfy quant charact equ quant counterpart spect algebra curv op coordin construct provid loc sep vary condit enl necess glob sep vary tak plac
giv proof dilogarithm id assocy minim model virasoro algebr
investig glob property map ent describ symmetry integr spin vertex model exploit nat bir transform project spac giv algorithm analys structure inv map discuss charact condit quas integr particul link singul plan fin describ property dynam system mak contact arnol not complex exempl remark behavio
der quas quant group dimend topolog field the direct class act path integr detail comput carry chern simon the finit gaug group principl behind comput presum gen extend class act dimend topolog the manifold dimend less construct gen path integr dimend reduc standard dimend reproduc quant hilbert spac dimend topolog the path integr circ categ repres quas quant group pap consid finit the gen path integr reduc finit idea nee extend beyond finit the tre
prov id betweenth valu rog dilogarithm funct describ connect id spectr conform field the
equ gen restrict solid solid rso model assocy set graph mult colo loop model est apply consid solv loop model obtain solv famy crit rso model famy class dynkin diagram simply lac algebra rso model label algebr pair rel vertex model giv crit extend break symmetry dynkin diagram
construct four dimend topolog quant field the modul tens categ complet proof cas root un construct import phys interpret chern simon stat ashtek vary
giv princip bundl smoo manifold compact group giv finit dimend unit repres defin algebr funct spac connect modulo gaug transform holonom banach algebr complet algebr gen regul wilson loop el dual regard substitut meas spac connect modulo gaug transform nat linear diffeomorph inv el spac complex valu amby isotop inv fram ory link moreov simil result hold algebra analog holonom algebr result clar rel diffeomorph inv gaug the link inv fram depend expect valu produc wilson loop
form combin approach inv three dimend manifold reshetikhin turaev wit framework perturb topolog quant chern simon the correspond arbit compact simpl group pres direct impl surgery instruct context quant field the propos explicit form spec inv group der shown agr result lick subject class
star produc class doubl group simpl group correspond symplect grupoid giv quant doubl quant tang bundl obtain deform describ complex quant group bicov quant algebra discus point view discuss quant poisson structure symmet algebr lead quant envelop algebr exampl biqu sens turaev describ term gen bicov diff calcul conveny purpos fin interpret deform framework wel known property compact quant group simpl consequ correspond property class compact group analog class kirillov univers charact formul giv unit irreduc repres compact cas
outlin bas principl canon form nambu mech gen hamilton mech propos yoichiro nambu bas not nambu bracket gen poisson bracket multipl op high ord class observ describ hambu hamilton equ mot giv hamilton introduc funda id nambu bracket replac jacob id consist condit dynam show nambu structure giv ord defin famy subordin structures low ord includ poisson structure satisfy certain match condit introduc analog act principl least act nambu mech show dynam loop dimend object nat appear form discuss sev approach quant problem pres explicit repres nambu heisenberg commut rel cas emphas rol high ord algebra op mathem structures rel play pass hamilton nambu dynam pict
develop approach faddeev reshetikhin takhtas majid en assocy quasitriangul hopf algebr every regul invert const solv quant yang baxt equ show hopf algebr act quant doubl
reform method rec propos construct quasitriangul hopf algebra quant doubl typ mat obey yang baxt equ und algebra structures method elucid illust facil giv lat produc exampl quasitriangul hopf algebr correspond univers matrix pres form pow sery
braid diff op obtain differenty addit braid covect spac introduc prevy braid addit quant plan affy yang baxt matrix quant eigenfunct vecx vecv braid plan wav introduc fre cas posit compon tot commut prov braid binom theorem braid tayl theorem vec vecx vec vecx vary result prec gen gen matrix hent dimend wel known property us dimend diff expon rel apply show heisenberg algebr braid semidirect produc cocross braid lin act braid weyl algebr simil gen arbit matrix
gen matrix typ stat model link inv construct contain matrix inv gen funct univers vassiliev link inv express simpl kontsevich express quant defin combin contain integr exceiv express univers drinfeld assocy
gen standard homolog techn algebra comput sem infinit cohomolog algebr vary mod comput provid phys stat grav coupl minim model fre scal field
topolog inv clos connect ory four dimend manifold propos inv construct surgery spec link four dimend counterpart celebr three manifold inv reshetikhin turaev wit
study stringy gen partit funct scft shown comput us anoma decoupl brst triv stat partit funct particul limit partit funct yield partit funct topolog the coupl topolog grav apply comput numb holomorph ellipt curv certain calab manifold includ quint threefold view first apply mir symmetry string quant level
quant bialgebr rel baxt eight vertex matrix found quant deform algebr valu automorph funct complex tor
trigonomet deg sklyanin algebr found funct real repres spac polynom vary study contract giv standard quant algebr shown deg sklyanin algebr contain subalgebr isomorph algebr funct quant sphere diagon gen quadr form gen lead funct real diff equ askey wilson polynom
class algebra high dimend rec construct let show finit gen subalgebr isomorph algebr loc diffeomorph dimend moreov tow infinit many field transform subalgebr symmet tens dens also unravel structure isomorph schouten symmet bracket provid nat gen infty high dimend
reexamin infty symmetry conform affin tod the shown poss reduc nonun zero curv equ equ first ord pseudodiff opr whos coefficy gen infty algebr clar known rel conform affin tod the hierarchy poss correspond matrix model conform affin tod model discuss
pap gen result loop algebra group wel virasoro algebr dimend cas defin study class infinit dimend complex group cent extend group smoo map dimend ory surfac without bound simpl complex group extend nat correspond complex curv kernel extend jacob curv study coadjoint act show orbit label modul holomorph princip bundl curv describ langu part diff equ gen also poss describ orbit conjug class twist loop group lead consid diff equ holomorph funct giv ris hop describ group possess counterpart rich repres the develop loop group also defin dimend analog virasoro algebr assocy complex curv gen study complex analog hil op yield describ inv coadjoint act algebr answ turn dimend inv coincid extend algebr cur
show integr enjoy ex infinit many abel conserv charg satisfy yang baxt symmetry algebr charg gen quant monodrom op provid repres deform affin algebra review gen work veg eichenher maillet bootstrap construct quant monodrom op sin gordon mass thir model op possess class analog within light con approach model explicit comput eigenvalu vertex altern transf matrix gen phys stat algebra beth ansatz thermodynam limit turn valu period funct determin gen loc abel charg includ energy moment yield abel subalgebr loc algebr particul bootstrap result coincid ratio determin lat transf matrix
explicit formula univers matrix giv quant nontw rank affin algebra
denot quant affin algebr quant nontw affin algebr categ defin sect show deform paramet root unit integr repres categ complet reduc every integr irreduc highest weight mod correspond equ unit mod
calc larg asymptot expand exact surgery formul wit inv seifert manifold contribut flat connect ident agr loop formul check contribut irreduc connect appear contain finit numb term asymptot sery loop correct contribut triv connect found proport casson inv
study subdivid property certain lat gaug the bas group four dimend boltzman weight shown inv typ subdivid mov certain discret valu coupl paramet partit funct provid combin inv und simpl complex least bound also show extr phas fact ar comp boltzman weight alexand mov bound undergo subdivid
rec rog dilogarithm id attract much at set conform field the wel lat model calc connect threads id richmond szek appear comput cent charg conform field the show richmond szek id extend kirillov reshetikhin interpret lift gen third integr homolog finit cyc subgroup sit insid project spec linear group tim real mat view discret group connect allow clar assert conject stat work nahm recknagel terhov concern rol algebra the thurston program hyperbol manifold spec rel hyperbol manifold suggest appropry rel group manifold univers cov group project spec linear group tim real mat view topolog group also resolv weak vert conject form kirillov sum numb op conject mathem sid
gen deform heisenberg algebr introduc deform op instead gen many vary permit simult ex coh stat transposit cre op
defin mult colo gen braid monoid algebra pres explicit matrix repres rel dimend exact solv lat model stat mech particul show colo braid monoid algebr describ yang baxt algebr crit dilut model rec introduc warna nienhu seaton wel roch solv model rel dens dilut loop model discuss detail shown solv direct consequ algebra structure conject yang baxt gen mult colo braid monoid algebra lead construct solv lat model
consid class calab compact construct complet intersect weight project spac manifold ahl modul construct mir manifold calc instanton
heisenberg doubl hopf algebr regard quant analog cotang bundl group quant dual principl describ rel hopf algebr dual heisenberg doubl extend the coadjoint orbit class foury transform also describ twist heisenberg doubl relev study nont deform quant univers envelop algebra
describ mir orbifold repres class gen calab manifold real manifold dimend fiv sev despit dimend correspond superconform the perfect good compact heterot string four dimend spac tim check mir symmetry comput structure spac complex structures mir check reproduc known result yukaw coupl met appropry kahl class paramet orbifold togeth instanton correct
level spac distribut ar resc laguer jacob ensembl hermit mat study distribut express term fredholm determin integr op whos kernel express term bessel funct ord alph der system part diff equ assocy logarithm der fredholm determin und domain un interv cas singl interv fredholm determin painlev funct
show clebsch gord coefficy algebr depend singl paramet sqrt cont explicit calc smirnov wehrhahn phy
prevy pap us superspac techn prov perturb the around class solv zero mod chern simon quant field the gen manifold finit conject prov cas loop ad counterterm expect form term perturb the defin topolog inv pap prov conject proof us geomet compact reg feynm integrand feynm diagram smoo wel extend bas prop the
ellipt diagon solv reflect mat assocy ellipt matrix eight vertex fre ferm model pres lead second der op chain transf matrix hamilton magnet field inv quant deform clifford hopf algebr
mod affin superalgebra study particul shown study verm mod much result moody algebr gen sup cas import gen kazhd formul malikov feigin fuch construct giv weight explicit form singul vect verm mod affin moody superalgebra also consid decomposit admiss repres otim virasoro algebr modul transform tor fus rul diff bound condit tor correspond diff mod cur superalgebr charact sup charact might relev hamilton reduc result neveu schwarz ramond superconform algebra fin feld brst complex consist wakimoto mod fre field real construct
quant affin algebr prov univers matrix satisfy celebr conjug rel dag us twist apply braid gen shown diagon arbit tens produc mod integr irreduc highest weight mod spect decomposit formul braid gen obtain gen reshetikhin gould form pres affin cas casimir inv construct eigenvalu comput mean spect decomposit formul produc interest id found
revid contain reword select text
method obtain explicit complet topolog solv chern simon the develop effect necess aspect the colo ory braid dual property conform block correl wess zumino conform field the pres larg class repres gen groupoid colo ory braid obtain provid whol link inv jon polynom simplest exampl inv explicit calc illust knot upto eight cross compon multicolo link upto sev cross
not meas spac connect modulo gaug transform inv diffeomorph bas manifold import vary context mathem phys topolog form level exampl meas giv chern simon path integr certain meas sort also play rol stat quant grav ashtek form meas defin link inv gen multiloop inv not wit chern simon path integr giv ris jon polynom quant grav observ bas loop repres rovell smolin review rec work mak idea mathem rig giv rig construct diffeomorph inv meas spac connect modulo gaug transform gen rec work ashtek lewandowsk construct process lat gaug the graph analys embed bas manifold
gen formul phys observ chern simon the arbit compact group arbit clos ory three dimend manifold der term vacu expect valu wilson loop surgery pres kirby mov impl main ingredy approach cas explicit calc
construct class topolog lat field the three dimend defin gen class loc lat field the impos inv arbit topolog preserv deform und lat gen loc lat mov inv solv correspond hopf algebra satisfy certain constraint exampl study detail topolog lat field the correspond hopf algebr bas group ring show equ lat gaug the zero coupl ponzano reg the
form bas property vertex op context andrew baxt forrest sery exampl fac interact model der diff equ satisfy correl funct est connect repres the also discuss diff equ kashiwar miw sery exampl edg interact model next is model simplest spec cas sery study detail us jord wign ferm particul matrix el vertex op calc
explicit der el repres ring nee impl four dimend kirby calcul sketch
quant rham complex quant plan quant group construct zakrewsk deform produc deform dimend heisenberg algeb construct us construct model deform quant mech
known tetrahedron equ lead commut famy transf mat provid integr correspond three dimend lat model pres mod vert equ giv commut famy comply lay transf mat stat limit success construct solv equ term ellipt funct
construct quant group gaug the nee algebr inv gaug transform ex inv algebr clos rel ex diff algebr delt compat hopf algebr structure shown delt ex quant group quant group quant gaug group allow repres algebr delt structed op correspond diff real der spac irreduc repres help construct infinitesim gaug transform dimend class spac tim describ
repres quant group sem simpl algebr rank construct arbit repres rank quant group root un repres maxim dimend numb fre paramet irreduc repres ar spec cas
pres pap construct typ finit dimend repres quant superalgebr gen deform paramet deform cas finit dimend mod obtain irreduc decompos finit dimend irreduc opl submod
gen diff geometry form vect field cas quant algebra giv abstract form incorp many ex exampl diff geometry quant group combin extery der in der der form funct algebr particul find gen cart id hold whol quant univers envelop algebr left inv vect field implicit commut rel left inv bas form
method develop construct irreduc repres irrep quant supergroup system fash shown every finit dimend irrep quant supergroup gen deform finit dimend irrep und superalgebr ess un charact highest weight charact irrep giv root un irrep finit dimend multiply atyp highest weight irrep sem cyc irrep also ex exampl highest weight sem cyc irrep thorough study
orthogon polynom random matrix model hermit mat lead fredholm determin integr op kernel form pap concern fredholm determin integr op kernel form und un op interv emphas determin thought funct point interv show fredholm determin kernel gen form describ express term solv system long satisfy certain typ differenty formul also expon vary analys includ circul ensembl unit mat
examin problem determin repres braid group rieman surfac carry wav funct quant abel chern simon the interact dynam mat gen quant chern simon the cas coefficy chern simon term rat rieman surfac arbit gen tot mat charg van find explicit solv schr od equ find wav funct carry repres braid group wel project repres discret group larg gaug transform find funda constraint rel charg partic coefficy gen manifold
gen solv fact equ reflect mat thet eight vertex vertex model chain found assocy integr magnet hamilton explicit der find famy dependig sev continu wel discret paramet
rec wit propos consid ellipt gen superconform field the understand rel minim model landau ginzburg the pap first discuss bas property satisfy ellipt gener the property confirm funda class exampl introduc gen proc comput ellipt gener particul class orbifold the on orbifold neveu schwarz sect en calc ellipt gener landau ginzburg orbifold landau ginzburg orbifold allow interpret target manifold holonom comp express on obtain orbifold tens produc minim model also giv sigm model express ellipt gener manifold holonom
first part investig concern form grav framework uniform the rieman surfac first step direct show class liouvil act appear express correl topolog grav next der ineq involv cutoff grav background geometry anoth result alway rel uniform the concern rel high gen norm ord liouvil act furtherm show chir split anoma equ krichev novikov cocyc mean invers uniform giv real virasoro algebr arbit rieman surfac find eigenfunct holomorph cov op defin high ord cocyc anoma rel algebra fin attack problem consid posit sigm sigm liouvil field propos explicit construct foury mod compact rieman surfac
consid hierarchy nat typ hamilton system degr freedom polynom pot sep gen ellipsoid gen paraboloid coordin giv repres term tim mat whol hierarchy construct assocy linear matrix algebr matrix depend dynam vary yang baxt equ dynam typ propos us method vary sep provid integr system class mech conctruct sep equ hent explicit form act vary quant problem discuss help sep vary
us prevy obtain univers matrix quant nontw affin algebra determin explicit spect depend univers matrix correspond quant algebra apply reproduc wel known result funda repres also der extream explicit formul spect depend matrix adjoint repres simplest triv cas tens produc decomposit repres finit multipl
pap extend vert prevy short let cit attempt giv detail account result pres pap quant nontw affin algebr correspond quant simpl algebr us prevy obtain univers matrix determin explicit spect depend univers matrix apply spect depend univers matrix concret repres reproduc wel known result funda repres also abl der first tim extream explicit compact formul spect depend matrix adjoint repres simplest nont cas tens produc repres multipl fre
ord construct invers map period map primit form sem univers deform simpl ellipt singul saito introduc flat structure extend affin root system sect construct explicit flat thet inv cas typ us jacob form introduc wirthm ul combin result kato noum explicit describ flat coordin giv answ jacob invert problem linear isomorph period map simpl ellipt singul typ tild also detail publ elsewh
argu algebr gen pair deform boson allow hopfalgebr structure show imposs defin comultiply ev us nondeform cas ind comultiply us ord construct repres deform necess hopf algebra tens produc fock spac
rec dilogarithm id mad appear phys lit id seem allow calc structure const lik particul effect cent charg certain conform field the fus rul nahm recknagel terhoev proof id typ giv consid asymptot charact funct cal rog ramanud form comp asymptot predict modul cov refin argu obtain gen connect quant dimend certain conform field the argu dilogarithm funct id quest infinit consist condit paramet rog ramanud typ partit modul cov
discret model tod volterr chain construct continu boson hierarchy main tool discret symmetry preserv hamilton structure continu model boson cur hierarchy assocy sit correspond chain success act discret symmetry
consid diff typ deform linear group correspond us gen diagon matrix rel braid quant deform algebra coact quant plan discuss show tens grad preserv diff calcul construct braid group quant group quant plan cas diagon matrix
comput map class group act cyc config spac tor punct coefficy loc system ar conform field the act commut topolog act quant group giv vertex form
integr partic calogero mos system ellipt pot shown op found krichev possess class matrix structure matrix nat gen matrix found rec av talon trigonomet pot matrix depend spect paramet half dynam vary partic coordin satisfy gen yang baxt equ involv anoth dynam matrix
introduc assocy glu op opl spac solv quant yang baxt equ heck typ correspond glu op assocy quant group quant vect spac also found form involv tim quant mat whos entry squ rectangul quant mat correspond glu op link inv introduc involv stat model boltzman weight determin link inv glu standard solv assocy quant matrix group quant spac link inv ar rep glu dimend cas
study mean mir symmetry quant geometry ahl class paramet numb calab manifold main interest lie structure modul spac loc correspond singul model structure consid rich paramet vary paramet model study hitherto describ intrins structure point compact modul spac correspond larg complex structure class limit instanton expand interest ow fact instanton belong famy continu paramet comput yukaw coupl expand term instanton gen zero mak rec result bershadsky comput also instanton numb instanton gen particul valu paramet model becom bir certain model paramet compact div modul spac thu contain cop modul spac paramet model discuss process particul model anoth exampl somewh diff subject comp pap
construct infinit set loc conserv charg conform affin tod model techn involv abel dimend gaug pot satisfy zero curv form equ mot find infinit set chir charg apart lowest spin charg remain on possess chir dens charg diff chir poisson commut among discuss algebra property charg funda poisson bracket rel show charg conserv tim involv connect tod model est tak particul limit
mir symmetry picard fuch equ instanton correct yukaw coupl discuss within framework tor geometry allow est mir symmetry calab spac mir manifold unavail prevy construct mir map yukaw coupl explicit giv sev exampl three modul
loc macro includ goth fleqn
introduc addit us quant mat mean coaddit underlin delt otim otim suppl us comultiply delt otim togeth obey codistribut condit coaddit form us hopf algebr braid remark apply rectangul tim quant mat apply construct left inv vect field quant spac clos form braid algebr anoth apply wav funct lat approxim moody algebra lat field ad funct differenty
the vertex algebra constitut mathem rig axiom form algebra origin conform field the context borcherd algebra ar certain phys subspac vertex algebra review giv pedagog introduc rapid develop are mathem bas machinery form calcul pres axiom definit vertex algebra discuss connect conform field the der import imply axiom particul many explicit calc pres stress emin rol jacob id axiom vertex algebra class concret exampl vertex algebra assocy ev lat construct shown detail affin algebra fak monst algebr nat appear lead abstract definit borcherd algebra gen moody algebra bas property fin result simplest gen borcherd algebra analys point view symmetry quant the construct monst algebr sketch
class posit energy repres finit deg algebr infty construct term repres the algebr hatgl infty infinit mat finit numb zero diagon algebr unit on class wel simil result obtain algebra
nest beth ansatz gen op bound condit us find exact eigenvect eigenvalu vertex model fix op bound condit correspond inv hamilton beth ansatz equ obtain solv thermodynam limit giv vertex model fre energy hamilton ground stat energy includ correspond bound contribut
investig sev sphere group lik manifold extend moody lik algebr cov property tens composit spin defin rel malcev algebra est consequ octon project spac examin cur algebra form anoma der shown un ev regard num coefficy redefinit cur nilpot brst op consist particul express class field depend anoma sugawar construct giv
class matrix structure partic calogero mos model typ ellipt pot investig show moment independ matrix without spect paramet assum moment independ sufficy reproduc dynam mat av talon typ deg ellipt pot includ spect paramet en find mat gen ellipt pot
express defin rel deform minkowsk spac algebr wel correspond der diff form reflect equ form encompass cov property respect quant lorentz group act straightforward
provid explicit formul inv manifold introduc cran yet consequ result ex combin formul sign manifold term loc dat triang pot phys apply result ex light fact cran yet inv rig vert idea oogur wedg the
first part sery pap develop tens produc the mod vertex op algebr goal the construct vertex tens categ structure categ mod suit vertex op algebr not vertex tens categ ess complex analog not symmet tens categ fact vertex tens categ produc braid tens categ nat the apply particul many famili rat vertex op algebra includ assocy wznw model minim model moonshin mod pap part introduc not tens produc spec el modul spac spheres punct loc coordin pres funda property construct tens produc
analys modul spac calab threefolds assocy conform inv nonlinear sigm model show describ unexpect rich geomet structure spec kahl sect modul spac calab conform the admit decomposit adjac domain correspond complex kahl con topolog distinct manifold domain sep wal correspond singul calab spac spacetim met deg certain reg show un domain isomorph complex structure modul spac singl topolog calab spac mir resolv puzzl mir symmetry rais app asymmetry kahl complex structure modul spac calab manifold furtherm us mir symmetry show interpol phys smoo man the repres distinct point kahl modul spac ev point correspond topolog distinct spac spacetim topolog chang string the theref real bas op deform tru margin op fin work also yield import insight nat orbifold string the
rec work robert shown first surg manifold inv brod unspec norm fact stat inv ar tqft cran yet equ sign manifold subsequ brod defin anoth surg inv handl tre diff refin robert techn develop auth show improv surg inv brod also depend sign eul charact
ident examin gen topolog sigm model suit coupl topolog op strings target kahl manifold real structure involv act complex conjug compat kahl met model satisfy axiom might cal equ topolog quant field the gen axiom topolog field the giv atiyah observ equ topolog sigm model correspond cohomolog class equ cohomolog the target correl funct comput lead intersect the instanton modul spac nat real structure equ tim model discuss detail solv explicit fin discuss equ form topolog grav surfac un op clos string the find anoma explain problem form topolog op string the
gen compact ahl manifold shown toeplitz quant geomet quant lead wel defin op norm estim class limit gen ear result auth klimek lesniewsk obtain tor high gen rieman surfac respect thereby ar approxim poisson algebr sequ finit dimend matrix algebra infty
extend artin braid group op gen doubl tripl intersect consid extend alexand theorem rel intersect clos braid intersect knot prov doubl tripl intersect count exampl giv cas quadrupl intersect intersect knot inv construct markov trac defin intersect braid algebr repres extend turaev repres discuss exampl poss apply form quant grav discuss
second part sery pap pres the tens produc mod categ vertex op algebr part not tens produc mod vertex op algebr introduc certain hypothes construct tens produc giv us certain result stat without proof part proof result supply
defin calc fus algebr model rat level cohomolog method byproduc obtain cohomolog charact admiss repres wideh gtsl
homolog construct integr mot class quant tod field the giv us construct ident integr mot cohomolog class certain complex model resolv assocy algebra quant deform prov class integr mot quant tod field the assocy finit dimend algebra algebr integr mot correspond algebr affin tod field the algebr commut subalgebr algebr consist quant hamilton
lipm ber univers teichm ul spac class denot play sign rol teichm ul the teichm ul spac fuchs group embed complex submanifold rec also becom object intend study phys prom geomet environ perturb vert boson string the provid techn survey cur known geometry conject phys mean
quant gen rog fiv term pentagon dilogarithm id suggest shown class limit giv us rog id cas quant id real finit dimend spac also consid quant dilogarithm construct funct ferm curv id equ restrict star triangl rel introduc bazhanov baxt
describ harmon superspac form wit manin supertw correspond extend sup yang mil the ess sufficy supersymmet extend yang mil equ mot recast form cauchy rieman lik holomorph condit pair prepot appropry harmon superspac form mak explicit construct solv rath tract proposit prevy attempt
consid correl funct wess zumino wit model tor insert cart el mathem mean consid funct form ldot intertwin verm mod evalu mod affin algebr ghat grad op verm mod cart subalgebr der system diff equ satisfy funct particul calc frac yield parabol second ord clos rel heat equ compact group correspond consid detail cas sltwo cas follow diff equ left frac frac right frac crit level becom equ cas der integr formula find asymptot thu recov class funct
vertex op assocy level wideh mod construct explicit us boson ferm integr formul der trac produc vertex op result apply giv point spin correl funct integr quant spin chain extend ear work jimbo cas
certain gen algebr first ord diff op act spac inhomog polynom construct gen algebr depend permut op shown hamilton body calogero model repres second ord polynom gen algebr giv repres imply calogero hamilton possess infinit many finit dimend inv subspac explicit bas clos rel finit dimend repres algebr repres altern standard repres bargman fock typ term cre annihil op
glob model deform quas orthogon algebra gen group mot four dimend affin cayley klein geometry obtain start three dimend deform shown main algebra class property system impl quant cas quant deform vert eith spac tim spac symmetry algebra poinc galile euclid wel oth appear context particul cas sev deform direct obtain
pap study monoid categ dual tens produc nee commut mot exampl categ repres hopf algebra mot apply definit symbol us topolog field the introduc not spherical categ first sect prov coh theorem monoid categ dual follow mac second sect giv definit spherical categ construct nat quoty also spherical third sect defin spherical hopf algebra categ repres spherical exampl spherical hopf algebra involv hopf algebra ribbon hopf algebra fin study nat quoty cas show semisimpl
princip bundl us techn loop repres gaug the construct wel defin substitut lebesgu meas spac connect haar meas group gaug transform prec defin algebra cylind funct spac defin gen meas spac continu linear funct correspond algebra borrow idea lat gaug the charact gen meas term graph embed charact construct gen meas respect uniform gen meas inv group automorph project gen meas consid ashtek lewandowsk cas gen haar meas right left inv wel inv show av gen meas gen haar meas giv gaug inv gen meas
mean tor geometry study hypersurfac weight project spac dimend four particul comput giv manifold intrins topolog coupl find result agr calc correspond coupl mir model larg complex structure limit
infinit dimend quant affin algebr famy cent el casimir inv construct eigenvalu comput integr irreduc highest weight repres eigenvalu formula shown absolv converg deform paramet prov univers matrix satisfy celebr conjug rel dag us twist apply braid gen shown diagon arbit tens produc mod integr irreduc highest weight mod spect decomposit formul braid gen obtain gen reshetikhin gould form pres affin cas casimir inv act spec mod also construct eigenvalu absolv converg comput mean spect decomposit formul
analys glob aspect modul spac ahl form conform sigm model us algebra method mir symmetry study extend mathem not leng spec ahl structure conform field the calc length chang modul field vary along distinct path modul spac find strong evid support not robust set quant calab modul spac string the restrict poss ahl form enforc minim leng scal provid topolog chang prop tak account length howev shrink zero also comp stringy geometry class gen rel context
prop account inv calab sigm model shift field integr amount an thet angl show modul spac sigm model oft enlarg includ larg radi limit point simplest cas holomorph coordin enlarg modul spac van limit point appear multiply front instanton contribut yukaw coupl instanton contribut theref suppress limit point comply cas instanton contribut stil suppress enlarg spac singul limit point singul interest effect effect four dimend the calab us compact heterot string
not correspond rath acc transl lect giv fif mex school partic field held guanajuato decemb constitut brief el introduc quant symmetry phys point view along lin forthcom book omez sierr
consid modul spac flat connect rieman surfac mark point efficy paramet suggest us construct integr model modul spac famy commut hamilton extract trac transf matrix built wilson lin observ chern simon the model appear gaug equ magnet chain finit numb sit
pair univers envelop algebr algebr funct obtain explicit term regul repres quant doubl construct investig structure root subspac casimir op rev describ term el
mot suggest paolo cott ramusino work phys level rig rel the donaldson polynom provid construct appl turaev viro cran yet inv prior fin inv depend cho homolog class manifold
fedosov giv simpl nat construct deform quant symplect manifold us flat connect bundl form weyl algebra assocy tang bundl symplect manifold connect obtain affin nonlinear it flat giv tors fre symplect connect pap class analog fedosov op connect analys shown produc us expon map linear connect ordin manifold symplect vert also analys fin remark mad imply deform quant fedosov index theorem gen symplect manifold
introduc study kosz complex heck matrix cohomolog cal berezin us defin quant superdetermin heck matrix behavio respect heck mat study giv heck matrix dimend vect spac construct heck matrix dimend vect spac commut diff not quant diff supergroup der algebr funct diff coquasitriangul hopf algebr us algebr diff form quoty exampl superdetermin rel algebra calc sev remark woronowicz the mad
show extern algebr equip grad poisson bracket compat group act prov grad poisson structures bracket obtain explicit describ real structures appear quasiclass limit bicov diff calcul quant linear group
show every algebr sup algebr canon braid term envelop algebr appear flat spac braid commut coordin funct also giv point view minkowsk spac ar simil envelop algebr braid algebr point view fix sign met minkowsk spac hent also ordin minkowsk spac also describ abstract construct left inv integr braid group
consid poss cov extern algebr structures cart form bas upon follow nat post inv form real adjoint repres quant group monom form possess un ord obtain extern algebra defin extery der possess us nilpot condit gen deform vert leibn rul stat known exampl diff calcul propos class scheme problem reduc discuss
conceiv univers matrix rec introduc fronsd galindo framework quant group discuss gen expon map exampl rel inhomog quant group phys interest develop dual calc explicit pres found cas univers matrix lik group express term us expon sery
pap quant integr hamilton quant many body problem interact pot sinh sutherland op construct im high casimir algebr certain homomorph cent algebr diff op vary simil construct apply affin crit level defin correspond high sugawar op quant integr hamilton quant many body problem pot eq const tim weierstrass funct allow giv proof olshanetsky perelomov theorem stat hamilton defin complet integr quant system also giv express eigenfunct quant integr sutherland op trac intertwin op certain repres
pap develop the construct inv clos ory manifold giv certain typ hopf algebr exampl giv quant envelop algebr semisimpl algebr semisimpl involv hopf algebr inv defin stat model triang cas inv partit funct topolog quant field the
pres pap giv diff geometry form bas dynam principl group algebra approach cit grad condit term holomorph distribut flag manifold assocy parabol subgroup complex group der correspond nonlinear integr system gen solv moreov real condit solv introduc cas simpl group endow canon grad system quest reduc abel tod equ obtain gen uck repres pseudo met spec ahl met flag manifold rel maxim nonsemisimpl parabol subgroup gen infinitesim uck formula ricc curv tens pseudo met accord formula funda form pseudo met ricc curv tens express direct term abel tod field sens ahl pot
introduc work gen quant group mackey approach quant homog spac recal bicrossproduc model auth gen quant doubl describ gen extend the hopf algebra nonabel cohomolog spac class form kind topolog quant numb phys vis quant world cross produc quant also view triv quant princip bundl quant group gaug the also rel nonabel cohomolog drinfeld the twist
consid canon symplect structure modul spac flat connect rieman surfac gen mark point semisimpl algebr obtain explicit efficy formul symplect form prov repres cop kirillov symplect form orbit dress transform poisson group cop symplect structure heisenberg doubl poisson group pair correspond algebr
study repres the finit algebra introduc parabol subalgebra describ structure algebra defin verm mod giv conject determin allow find complet deg repres finit algebra extract irreduc repres analys structure singul subsingul vect find algebra gen maxim submod verm mod gen singul vect surpr rol singul vect encaps term dual analog kazhd lusztig theorem simpl algebra involv dual rel kazhd lusztig polynom support conject exampl brief discuss apply gen infinit algebra
class dynam mat body calogero mos model ellipt pot giv body cas interest nont structure goe beyond known ansatz moment independ mat mat pres includ dynam mat av talon deg cas ellipt pot
pres pap describ geometry abel finit period conform tod system assocy sery simpl algebra endow canon grad princip tool gen class uck embed cas flag manifold assocy funda repres direct proof correspond ahl pot satisfy system dimend finit period conform tod equ shown geometry typ ment coincid diff geometry spec holomorph surfac target spac submanifold quadr appropry cho addit surfac defin satisfy quadr holomorph diff condit ens consist gen uck embed condit autom fulfil tod equ hold
construct matrix gen calogero mos system introduc gibbon herms reduc proc obtain matrix eul calogero mos model standard calogero mos model
top thes develop versatil geomet mot diff calcul commut quant spac provid pow easy mathem tool apply phys rel sci gen unit tim evolv propos study simpl level system lead conserv microscop entrop phenomenon quant mech cart calcul combin funct form der in der along gen vect field algebr construct quant group extend quant plan construct tang bundl quant group manifold brst typ approach quant group gaug the giv exampl apply mat org part part study vect field quant group emphas hopf algebra structures also introduc quant geomet construct us gen sem direct produc construct combin dual hopf algebra funct left inv vect field ful bicov algebr diff op pur braid group introduc commut delt provid inv map thereby bicov vect field casimir met construct allow transl undeform matrix express less obvy quant algebra count part study detail quasitriangul hopf algebra giv determin
cart calcul der diff form in der bas undeform cart id construct attempt class vary typ quant algebra pres fair gen exampl construct util pur braid method prov orthogon adjoint repres giv kil met quadr casimir reform cart calcul braid algebr extend quant plan direct induc group calcul provid
scal funct determin sect dimend disc spherical determin arbit angl rat fact wholesph hemisph express also giv dimend ordin conform laplac comparison mad work
discuss rel vaf quant cohomolog flo homolog the introduc equ quant cohomolog form conject gen property bas conject comput quant cohomolog algebra flag manifold answ turn coincid algebr regul funct inv lagrang vary tod lat
repres theoret approach spec funct develop work gelfand naimark vilenkin collab ess approach fact class spec funct obtain suit spec matrix el charact repres group anoth rich sourc spec funct the clebsch gord coefficy describ geomet juxtaposit irreduc compon insid tens produc repres fin rec work repres quant affin algebra shown matrix el intertwin op certain repres algebra interest spec funct hypergeomet funct gen pap suggest gen method get spec funct repres the un three method ment allow defin study many spec funct illust method numb exampl macdonald polynom eigenfunct sutherland op lam funct
pap pres formul macdonald polynom root system ar repres the quant formul express macdonald polynom weight trac intertwin op certain mod quant also describ commut system macdonald diff op us gen cent quant univers envelop algebr describ prov trac formul gen eigenfunct op funct gen hypergeomet funct rel solv quant knizhnik zamolodchikov equ
artic devot quant field the expl rel not year gepn verlind algebr group level cohomolog grassman argu process start dimend sigm model whos target spac grassman integr field standard long known result energy effect act describ the mass novel the fact equ long dist gaug model hent rel verlind algebr
property quant minkowsk spac algebr discuss irreduc repres highest weight vect construct rel quant algebra oscil sphere point
obtain matrix matrix repres artin braid group act canon vect spac every sup algebr braid algebr result apply every sup hopf algebr braid hopf algebr recov known repres assocy rack also obtain repres triv ring polynom vary regard braid lin repres extend artin braid group braid compl also obtain method
review rec develop dimend stat mech corn transf mat provid vit link phys system repres the quant affin algebra op many poss eigenst describ us property vertex op
show mult boson hierarchy possess class discret symmetry link discret tod system discret symmetry gen simil transform correspond op est canon nat discret transform spect equ defin lat system correspond op play rol determin pertin symmetry structure also introduc conceiv squ root lat lead famy pseudo diff op cov addit cklund transform
clar rel approach minkowsk spac carow watamur approach bas ide tim braid hermit mat lat object lik sup mat bos ferm stat replac braid stat also obtain matrix formula poinc group framework
construct cohomolog the control deform gen drinfel algebr pict pres sid combin rel fact ex grad algebr structure simpl cochain complex associahedr algebra rel algebr der construct
first review property conv funct tod lat hierarchy straightforward gen discuss correspond pass diff finit diff equ involv howev conceiv op valu funct assocy cart level algebra pres study could us understand bet fre field rel ordin fre field
repres braid group obtain rat conform field the us obtain explicit repres temperley lieb jon algebra method describ detail wess zumino conform field the gen arbit rat conform field the outlin explicit definit assocy linear trac op term certain matrix el spac conform block conform the pres every prim field rat conform field the subfact hyperfinit fact triv rel commut index subfact giv term id id el certain dual matrix conform block four point correl jon formul index subfact correspond spin frac repres wess zumino conform field the definit trac op also provid method obtain link inv explicit
pres pap devot vary object infinit dimend geometry second quant fre string purpos includ symmetry gen infinit dimend geomet pict rel quant field the strings describ first part pap algebra group geom appear don chang algebr infinitesim reparamet string world sheet quas pseudo algebr class transform gerva matsuo quas pseudo algebr wel virasoro algebr cent extend enlarg gerva matsuo quas pseudo algebr obtain algebra class transform gerva matsuo quas pseudo algebr propos rel gerva matsuo diff geometry batalin weinstein karasev maslov approach nonlinear poisson bracket wel sabinin program nonlinear geomet algebr ment
review diff method calc wit inv stat phas approxim surgery calcul giv detail describ loop approxim formul wit inv techn involv der exact valu surgery construct manifold fin comp formula produc method dimend sphere len spac
chern simon quant field the ord prov rec conject limit expand jon polynom knot rel alexand polynom limit allow der surgery formul loop correct contribut triv connect wit inv loop part formul coincid walk surgery formul casson walk inv prov conject casson walk inv loop correct triv connect contribut wit inv rat homolog sphere contribut triv connect wit inv manifold nont rat homolog calc cas seifert manifold
artic devot interest geomet inform interpret peculi quant field the becom veal vis cont geometry mobilevid interpret geometry anom virt real quant project field the mobilevid quant conform conform field the quant field analog eul arnold org cyberspac alexandr geometry mobilevid inform mobilevid inform transmit anom virt real photodosy inform transmit int anom virt real iavr teleaesthesy
shown principl loc noncommut geometry connnect sheaf theoret method framework quant spac introduc exampl mathem phys giv langu quant spac noncommut princip vect bundl defin property study import construct class the princip fibr bundl lik assocy bundl diff calcul carry quant cas deform instanton model introduc every integr index
upto cross numb knot whos chir detect known polynom nam jon inv vary gen homf kauffm inv show gen knot inv obtain chern simon topolog field the giv known polynom spec cas indee sensit chir knot
explicit isomorph est quasitriangul hopf algebra study rec ogievetsky pres auth
comput quant cohomolog ring part flag manifold cdot tim cdot tim induc comput us ide giv also defin not vert quant cohomolog ring algebra bundl flag bundl cdot assocy vect bundl ring found
study quant matrix algebr standard tim cas propos ordin deform euclid spac algebr simplest cas isomorph us quant mat form nat cov euclid rot otim also introduc quant wick rot twist system prec approach minkowsk spac bas braid mat assocy spin lorentz group
poss grad poisson structures extern algebr describ prov diff poisson structures prolong sklyanin bracket ex grad poisson structures neith obtain reduc grad poisson structures extern algebr quant result grad algebr quant right inv form three gen
act tot cohomolog almost kahl manifold flo cohomolog roduc origin flo giv ring structure prov tot cohomolog spac provid ring structure isomorph quant cohomolog ring spec cas prov formul flo cohomolog ring complex grassm conject vaf wit
commut diff calcul quant group extend introduc analog class cas in produc op der cas show extend calcul induc coact simil extend calcul cov quant plan in produc op der introduc plan wel situ quant group quant spac brief discuss explicit formula giv dimend quant plan
recurs method der calc eigenvalu correl funct random hermit matrix larg siz limit smooth short scal oscil property point funct univers recov three four point funct giv explicit observ high ord correl funct linear combin univers funct coefficy depend increas numb paramet matrix distribut
prov rec conject rel tim matrix diff op part certain linear loc poisson bracket algebr algebr contain virasoro subalgebr appear study abel tod field the show algebr prec giv second gelfand diki bracket assocy miur transform giv rel second first gelfand diki bracket gelfand diki bracket also obtain assocy integro diff equ satisfy ferm bilinear asymptot expand resolv study coefficy yield infinit sequ hamilton mut van poisson bracket recal lead matrix hierarchy flow equ three compon field reduc ordin hierarchy correspond matrix mkdv equ also giv wel rel pseudo diff op approach result continu hold hermit tim matrix conject mad tim matrix ord diff op assocy algebra
show belavin solv quant yang baxt equ obtain restrict infinit matrix suit finit dimend subspac infinit matrix mod vert shibukaw ueno matrix act funct vary
study repres map class group punct tor doubl finit dimend poss semisimpl hopf algebr ar construct univers extend topolog field the discuss doubl deg problem tqft circumv find compact formula mat us canon deg form hopf algebra bicross structure doubl rath monodrom mat rig proof modul rel comput project phas supply us radford rel canon form modul integr analys project act cent root un appear dimend repres decompos dimend finit repres dimend irreduc repres lat tens produc dimend standard repres finit dimend repres obtain trunt tqft semisimpl repres categ
study poss defin braid comultiply cov diff complex quant spac discov diff bialgebra hopf algebra boson ferm quant hyperpl addit coproduc braid matrix algebr multiply addit coproduc lat cas rel minkowsk spac poinc algebr
introduc algebra the integr quant plan braid spac dimend cas obtain novel pict jackson integr indefinit integr braid group funct vary tre braid stat rath us boson grassman on show definit integr also evalu algebra multipl integr gauss remain boson scal vary assocy deform compos algebra integr repres lead ordin numb integr also integr develop ful the foury transform braid addit delt otim otim braid antipod defin convolv produc prov convolv theorem prov also prov analog result braid group includ integr foury transform quant plan assocy gen mat includ euclid minkowsk spac
artic review duisterma heckm integr formul ensu equ cohomolog structure finit dimend cas particul discuss connect equ cohomolog class integr also explain integr formul der expl poss direct could ev yield novel integr formula nont integr model
correspond braid group handlebody arbit gen algebr yang baxt extend reflect equ op shown repres infinit dimend extend reflect equ algebr term direct produc quant algebr gen der lead repres braid group term mat restrict reflect equ op giv colo braid group reflect equ op describ effect handl attach bal satisfy charact equ giv ris addit skein rel thereby inv link handlebody origin skein rel explain der adequ adapt handlebody vert jon polynom relev result construct link polynom clos manifold heegard splitting surgery ind
comput poisson bracket rel monodrom matrix auxy linear problem bas poisson bracket rel model contain der comput lead peculi typ symmetry break account spont quant und glob gaug group class poss pattern symmetry break outlin
algebra form quant group cov noncommut geometry framework matrix approach the quant group giv consid structure group tak valu quant group introduc not noncommut connect curv transform comod loc coact structure group extery extend noncommut connect curv gen cov quant algebra algebra find combin gen inv coact loc quant group form consid inv noncommut im lagrang topolog chern simon model abel gaug the einstein grav pres also explicit real cov quant algebra investig coset construct otim
regul defin addit coproduc coaddit deform diff complex propos quant group quant spac rel heck typ mat sev exampl braid coaddit diff bialgebra hopf algebra pres
commut diff calcul quant group construct quant extern algebr propos contain numb gen class cas extery der defin construct obey mod vert leibnitz rul
dilut gen birm wenzl murakam algebr consid baxt solv yang baxt algebr vertex model exampl correspond solv lat model regard dilut vert vertex model
obtain nonperturb result framework continu liouvil the particul express spec heat pur grav term expand integr modul spac punct rieman spheres integrand writ term liouvil act show satisfy painlev
polynom inv correspond funda repres gaug group comput arbit tor knot link framework chern simon gaug the mak knot op result formul homf polynom arbit tor link pres
explicit express three sery mat rel dilut gen birm wenzl murakam pres sery equ quant mat gen tod system wherea remain sery appear
mot quest path integr approach topolog gaug the address follow quest giv smoo manifold compact group poss smooth diagon conjug maxim tor analys loc glob obstruct giv complet solv problem regul map est alway smooth diagon loc obstruct glob triv weyl group tor bundl show patch loc diagon map giv ris triv bundl explain rel wind numb map restrict structure group examin behavio gaug field diagon also discuss comply ar regul map pres triv bundl particul est rel ex regul sect triv adjoint bundl restrict structure group princip bundl result just weyl integr formul funct integr novel feat seen finit dimend cas contain sum topolog sect ar restrict triv princip bundl us prevy solv complet yang mil the model dimend
describ topolog string the reproduc many aspect expand yang mil the spacetim dimend zero coupl limit string the mod vert topolog grav coupl topolog sigm model spacetim target der string the rely interpret gross tayl omeg point describ includ are coupl chir sect wilson loop expect valu incorp topolog string approach
describ mir manifold dimend diff famili cas complex threefolds emphas simpl feat dimend three supply robust method rely spec charact hent nat gen dimend modul spac calab fold somewh diff spec ahl manifold occur ind geomet structures ar form apply proc allow construct mir map calc ord ord instanton correct yukaw coupl mathem correct expect correspond calc chern class vary paramet spac hilbert schemes rat curv calab manifold result agr obtain tradit mathem method limit numb cas lat analys carry fin mak explicit striking rel instanton correct vary yukaw coupl der assocy op produc algebr
short not devot set fre hid symmetry verm mod noncommut verones map
extend univers diff calcul arbit hopf algebr univers cart calcul accompl introduc in der der el univers diff envelop algebr form incorp object univers diff calcul togeth consist commut rel also explain includ nont commut rel form obtain gen cart calcul
explicit construct sery lat model bas upon gaug group property subdivid inv coupl paramet quant field config restrict satisfy typ flat condit simplest model typ yield dijkgraaf wit inv manifold bas upon singl link simplex field depend upon manifold dimend model specy field vary bas high dimend simpl
novel hopf algebr tild depend deform paramet fiv gen construct tild hopf algebr might consid quant class otim group contain standard quant group hopf subalgebr howev interest observ paramet deform quant group also real gen tild algebr provid set deform paramet rel particul fash subsequ construct inv noncommut plan assocy tild algebr show wel known manin plan correspond quant group easy reproduc construct fin consid colo extend quant group wel correspond manin plan expl intim connect colo extend tild hopf structure
pap devot mathem aspect topolog quant field the apply enum problem algebra geometry particul contain axiom tre gromov wit class discuss property fano vary cohomolog field the defin prov tre level the determin correl funct apply count rat curv pezzo surfac project spac giv
solv twist yang baxt equ ar intertwin cyc repres wideh describ coupl lat cur algebra
extend result prevy pap knot link us formul jon polynom link der rec reshetikhin illust formul exampl tor link rel paramet reshetikhin formul multivary alexand polynom est check express alexand polynom satisfy bas property fin der link surgery formul loop correct triv connect contribut wit inv rat homolog spheres
real hopf algebr algebr diff op quant euclid spac gen suit deform analog angul moment compon ordin algebr funct splits direct irreduc vect repres lat explicit construct highest weight repres
introduc integr tim discret vert class calogero mos model goe origin model continu limit discret model obtain pol solv discret vert kadomtsev petviashvil equ lead finit dimend symplect map pair symplect structure sufficy inv discret calogero mos model construct class matrix continu model
mot rec pap fock ros cit foro describ mathem prec quant hamilton chern simon the introduc chern simon the lat expect reproduc result continu the exact lat model enjoy symmetry respect quant gaug group us fact construct algebr observ hamilton chern simon the equip op posit in produc
conv discret approxim manifold preserv nont topolog feat artic describ approxim scheme sorkin reproduc phys import aspect manifold topolog striking fidel approxim topolog spac scheme part ord set poset ordin quant phys manifold continu prob dens gen commut algebr continu funct funda phys sign contain inform reconstruct topolog serv spec domain observ lik hamilton poset rol algebr assum noncommut algebr noncommut geometry bas noncommut algebra theref remark connect finit approxim quant phys noncommut geometry vary method quant phys us expl particul at paid develop num viabl approxim schemes tim preserv import topolog feat continu phys
show algebr bicov diff calcul quant group understood project cross produc braid hopf algebr quant doubl quant group result sup hopf algebr reproduc extend extery der tens produc
ellipt deform wideh propos pres algebr bas rel eight vertex mat ellipt modul chos diff trigonomet limit algebr reduc quoty propos reshetikhin semenov tian shansky conject concern highest weight mod vertex op form phys interpret discuss
import property hopf algebr quasitriangul us vary apply property investig quant group root shown diff form quant group root eith quasitriangul simil structure cal autoquasitriangul interest cas property mean braid automorph combin poisson transform adjoint transform certain el tens squ algebr
resolv baxt problem help automorph group yang baxt resp star triangl tetrahedron dot equ infinit group symmetry real linear bir coxet group act mat ex beyond narrow context strict integr yield among thing unexpect ellipt paramet integr sixteen vertex model provid class discret dynam system address rel problem charact complex it
pap study integr algebra integr property certain matrix schr od op spec assocy op rat trigonomet ellipt coefficy every simpl algebr every repres algebr nonzero finit dimend zero weight subspac calogero sutherland op spec cas construct op alway integr main result also algebra integr rat trigonomet cas repres highest weight gen correspond result calogero sutherland op prov chalyh vaselov also conject tru ellipt cas wel gen correspond conject chalyh vaselov calogero sutherland op
show diff complex omeg braid matrix algebr repres cov comod respect coact hopf algebr omeg diff extend hand algebr omeg cov braid comod respect coact braid hopf algebr omeg geomet aspect result discuss
giv gen construct finit dimend repres twist affin moody algebr der zero cent charg triv otim finit dimend algebr fact long hold quant deform cas necess tak direct sev irreduc mod form irreduc mod becom reduc illust techn work explicit exampl finit dimend mod determin multiplet structure soliton affin tod the
defin analog jack jacob polynom defin finit dimend root system heckm opdam eigenfunct trigonomet sutherland op affin root system affin cas defin polynom eigenfunct affin sutherland op calogero sutherland op ellipt pot plu term involv der respect modul paramet show polynom construct explicit trac certain intertwin affin algebr also defin analog construct giv affin analog macdonald polynom show conject rel macdonald in produc id affin cas scal produc conform block model
consid gen dunkl diff diff op assocy group gen reflect commut condit equ certain funct equ equ solv many cas particul solv assocy ellipt curv construct cas discuss rel ellipt calogero mos integr body problem discuss quant analog construct
describ detail spac ahl paramet calab manifold exploit mir symmetry larg complex structure limit mir correspond class larg radi limit found study monodrom period discrimin loc bound modul spac correspond singul calab manifold symplect bas period found act gen modul group determin mir comput instanton expand yukaw coupl gen index ar numb instanton gen zero gen degr also investig symmetry act bound modul spac
construct cocyc algebr pseudo pseudodiff symbol vary clos rel sin algebr poisson algebr tor quant godbillon cocyc pseudo diff op appear construct nat gen gelfand fuch cocyc period vect field describ nont embed virasoro algebr complet pseudodiff symbol propos analog hierarchy admit infinit numb conserv charg
exploit standard fact supersymmet yang mil the donaldson inv four manifold admit kahl met comput result agr avail mathem comput provid pow check standard claim supersymmet yang mil the
standard defin symplect hamilton structures first second gelfand diki bracket spac ordin linear diff op ord ldot pap consid detail cas tim matrix valu funct particul emphas interest second gelfand diki bracket particul interest reduc symplect submanifold reduc giv ris matrix gen class vert linear algebra cal algebra commut mat lead loc term algebra show algebra contain conform virasoro subalgebr combin form tim mat conform prim field spin analog scal cas gen howev algebra much rich structure algebra seen exampl linear loc poisson bracket matrix el work explicit matrix miur transform der map comply second gelfand diki bracket much simpl poisson bracket provid analog fre field real algebra
der quant deform dimend euclid poinc group zakrzewsk propos bas contract hopf algebr deform paramet sent construct kapp poinc deform algebr quant group obtain dual algebr
discuss not batalin vilkovisky algebr giv sev class exampl diff geometry the introduc not quant op algebr gen class op algebr exampl view deform commut algebr review not vertex op algebr show vertex op algebr addit structures fin est connect algebra voa
gen formul univers matrix quant nontw affin algebra khoroshkin tolstoy apply zero cent charg highest weight mod quant affin algebra shown univers matrix produc gauss decomposit trigonomit matrix tens produc mod explicit calc simplest cas pres consequ formula trigonomet matrix paramet tens produc verm mod obtain
embed method deform semisimpl algebra gen mot group dimend flat spac pres method giv glob simult scheme deform algebra on obtain wign contract dimend euclid poinc galile algebra
gen express bicov bracket gen extern algebr poisson group giv shown grad poisson structures der spec cas bracket formul univers apply cas matrix group
commut anoma schwinger term cur algebra dimend comput term wodzick residu pseudodiff op result writ twist rad cocyc algebr psdo construct second quant cur algebr clos rel geomet renorm interact hamilton gaug the
quant group structures spac unimodul mat quant group construct diff geometry tim bicov three gen satisfy liebnitz rul show diff geometry ex quant group der property
problem introduc depend el quant group class paramet consid suggest interpret homomorph algebr funct quant group algebr sect sheaf algebra class manifold describ depend argu funct point view group schemes appropry quant group field the sheaf hopf algebra manifold quant sheaf construct us boson formula algebr funct quant group the repr sent canon commut rel famy automorph hopf algebr depend class vary describ quant manifold manifold commut commut coordin discuss gen supermanifold quant group chir field rel algebra diff calcul discuss
purpos short not mayb consid com prov follow proposit proposit ex certain model mobilevid interpret fig observ mult us mod contrast singl us on result mayb interest apply understand theoret found quant interact process
modul spac string the target spac determin est particul discret symmetry group ful integr orthogon group ev unimodul lat sign method combin analys class the modul spac mir symmetry describ modul spac also pres viewpoint quant geometry consequ drawn concern mir symmetry algebra surfac
pap begin investig conceiv gen funct defin gen funct matrix el group el giv highest weight repres univers envelop algebr gen situ tim vary correspond el maxim nilpot subalgebra rath cart el moreov cas quant group funct numb tak valu commut algebra funct quant group despit diff particul cas conv funct integr tod lat hierarchy ar moody loop algebr level gen funct also satisfy bilinear hirot lik equ deduc manip intertwin op import apply form moody mult loop algebra pap contain illust calc simplest cas ordin loop algebr quant counterpart wel system funda repres
study inv dimend manifold bound depend bound condit best summ view exponenty inv el invers determin lin bound prov glu vary formul inv yield simpl proof holonom formul determin lin bundl famy dirac op also known glob anoma formul pap writ us amstex obtain am mathem socy instruct includ postscrib fil fig submit sep uuencod compress form
gen obtain larg spect paramet limit yang baxt op integr inv vertex model commut rel includ ser rel obtain limit yang baxt equ rec found eigenvect inv spin chain shown highest weight vect correspond quant group
pap contain attempt form rig check predict enum geometry curv follow mir symmetry main tool not stabl giv outlin contsruct gromov wit inv algebra project clos symplect manifold mir symmetry bas exampl rat curv quint fold reduc certain comply explicit id strategy comput describ follow reduc count problem quest concern chern class spac curv amby project spac us bot residu formul pass spac deg curv inv act group diagon mat tre evalu us techn feynm diagram comput scheme giv clos formula gen funct topolog sigm model wid class manifold cov many calab fano vary
three top attempt exply curios invers problem repres the op describ correct algebra object repres simpl exampl rel algebr consid mayb regard analog infinit dimend problem modern quant field the conform field the integr model field the triv background construct inspir top rel infinit dimend hid symmetry produc vertex op field discuss forthcom artic
gen prevy pap form residu formul simpl behavy equ quant cohomolog apply comput quant cohomolog part flag manifold cdot giv rig definit equ quant cohomolog
shown acklund transform zero curv repres zcrs system part diff equ pde clos rel connect est nonlinear repres symmetry group und induc gaug transform rel diff connect us construct zcrs vic vers furtherm proc outlin allow system search zcrs giv system pde
irreduc repres quant group woronowicz approach class wang parshal memoir cas root un find irreduc repres root un also ev degr wel describ diagon part tens produc irreduc repres exampl complet reduc repres giv ex haar funct prov correspond repres univers envelop algebra jimbo lusztig provid also recal cas gen comput don explicit
ex rigid calab manifold mir symmetry understood op spac manifold van first chern class artic continu investig particul typ ahl manifold posit first chern class gen calab manifold nat provid framework mir rigid string vacu class compr spec typ fano manifold encod cruc inform ground stat superst shown particul massless spectr supersymmet vacu cent charg crit der spec fano vary complex dimend crit certain circumst ev poss emb calab manifold high dimend spac construct describ lead insight rel exact solv model mean field the hand correspond calab manifold furtherm shown wit form landau ginzburg calab rel apply pres framework wel
analog tod lat hierarchy cal dispersionless tod hierarchy study dress op dispersionless hierarchy introduc canon transform quant dress op ordin tod hierarchy altern construct gen solv ordin tod hierarchy giv twist construct quat simil construct solv dispersionless hierarchy result wel obtain prevy pap pres proof necess techn detail
investig symplect geometry geomet quant class supermanifold exhibit brst structures certain kind algebra discuss undeform deform cas class wel quant cas
investig lat topolog field the defin chung fukum shap cont model defin tak deform quant doubl finit commut group und hopf algebr suggest chung fukum shap partit funct rel dijkgraaf wit zcfs rel hold
show kapp deform poinc quant algebr propos el partic phys structure hopf agebr bicrossproduc cobicross algebr semidirect produc class lorentz group act deform moment sect novel feat coalgebr also semidirect backreact moment sect lorentz rot us show kapp poinc act cov kapp minkowsk spac introduc turn necess deform commut also connect algebr prevy approach planck scal phys
reduc left regul repres quant algebr study diff intertwin op construct irreduc repres correspond spac loc sect certain lin bundl flag manifold
pres system techn construct solv yang baxt equ depend spect paramet addit continu paramet extr paramet ent yang baxt equ simil spect paramet addit form exploit fact quant compact algebra typ quant superalgebra known admit triv paramet famy infinit dimend finit dimend irrep respect ev gen develop techn construct correspond spect depend mat exampl work mat three quant algebra ment certain repres
construct repres quant algebr label complex numb act spac form pow sery commut vary vary gen flag manifold matrix quant group dual condit reduc proc construct diff intertwin op giv repres diff intertwin op giv explicit form
exampl finit dimend fact ribbon hopf algebr giv quoty quant univers envelop algebr root un degr map class group surfac gen hol project act automorph mod otim endow coadjoint mod structure ex project repres map class group surfac gen hol label finit dimend mod vect spac otim otim otim inv clos ory manifold construct mod construct class ribbon hopf algebra satisfy weak condit fact includ root un ev degr describ result mot
investig spin ferm quant spheres shown wav funct ferm dirac op quant spheres construct manifest cov quant group conceiv tot angul moment chir express us analog paul mat appropry commut rel
review vary construct mir symmetry term landau ginzburg orbifold arbit cent charg hypersurfac complet intersect tor vary particul shown diff techn rel
kahl modul spac particul simply connect calab manifold map us mir symmetry found model consid chir ring id diff assocy conform field the ambigu explain term model model langu also provid app counterexampl glob torell problem calab threefolds
pap deform gen linear group subject certain restrict particul ens smoo pass group limit obtain repres giv term certain set cre annihil op cre annihil op belong gen quark typ hadron typ boson ferm typ also nat definit direct sum algebra
extend discuss mir symmetry picard fuch equ instanton correct yukaw coupl topolog loop partit funct cas complet intersect high dimend modul spac develop method obtain instanton correct yukaw coupl clos study solv picard fuch equ lead clos formula prepot ahl modul field induc amby spac complet intersect singul weight project spac exampl tre part modul spac phenomenolog interest three gen model found class also apply method solv simplest model topolog chang observ discuss exampl complet intersect singul amby spac
asymptot solv quant knizhnik zamolodchikov equ assocy frak construct lead term asymptot solv beth vect eigenvect transf matrix quant spin chain model show norm beth vect eq produc hess suit funct explicit writ rat funct formul analog gaudin korepin formul norm beth vect shown gen beth vect form bas frak cas
pap study inv manifold der certain dimend hopf algebra inv bas right integr algebra shown result class inv distinct class wit reshetikhin turaev inv
investig equ gen mors the gen class integr model particul der equ vert class poinc hopf gauss bonnet chern theorem pres correspond path integr gen approach bas equ cohomolog loc techn clos rel form develop mattha quil approach gauss shap thom form
view knizhnik zamolodchikov equ mult tim nonautonom od equ transform heisenberg repres shown yield quant schlesinger equ quant form isomonodrom deform equ first ord op form rat tim matrix valu funct simpl pol matrix entry interpret op mod rat matrix loop algebr frak provid simpl form construct reshetikhin rel equ quant isomonodrom deform
pap meant inform introduc quant group start origin mot rec develop cal particul at new descov rel among quant group integr model jord structures
pap contain short review the symplect contact manifold gen the cas supermanifold shown gen us obtain import result quant field the particul regard superconform geometry particul cas contact complex geometry bet understand superconform field the connect topolog conform field the symplect geometry constitut mathem bas batalin vilkovisky proc quant gaug the exposit bas most publ pap howev pap contain also review unpubl result sect devot axiom superconform the topolog quant field the pap publ berezin mem volum
gaug inv not strong connect pres charact us just glob curv form defin strong connect interpret induc bas spac quant bundl exampl strong strong connect provid particul connect construct quant deform fibr certain class strong connect triv quant princip bundl shown equ class connect fre mod compat depend hermit met particul form yang mil act triv bundl investig prov coincid yang mil act construct con rieffel furtherm shown modul spac crit point act funct independ
pap ess short vert comput multiply anoma det detb ellipt pseudo diff op pdo clos manifold term symbol prov ellipt diff op clos posit definit on dimend introduc holomorph determin monodrom lie finit group root un gen cas rel multiply anoma cent extend group ellipt symbol inv quadr form extend comput algebr cent extend term logarithm symbol main tool trac typ funct defin class pdo integ ord canon ellipt pdo gen zet regul introduc
machinery braid geometry introduc prevy us construct epsilon tot antisymmet tens gen braid vect spac determin mat includ nat euclid minkowsk spac form complet cov correspond quant group widetild widetild hodg op diff also construct approach
charact der omeg univers property introduc class bimod
show construct cent grouplik quant determin algebra apply gen construct giv quant determin lorentz group
wish report rec approach commut calcul minkowsk spac bas reflect equ spect paramet consid express inv coact lorentz group commut property defin diff minkowsk algebra approach also allow discuss poss ambigu definit minkowsk spac diff calcul commut rel among gen coordin der lambd form inv scal op est comp ear result
study algebra structure doubl yang pres triangul decomposit fact canon pair yang dual insid consequ obtain explicit formul univers matrix demonst work evalu repres interpret dimend fact ar concret repres bilinear form highest weight polynom irreduc repres express form term gamm funct
lect giv pedagog expl not poisson structure extern algebr poisson group introduc prevy pap us not guid construct quant extern algebra prop class dimend
show matrix string model correspond coupl system dimend mod integr equ subject spec symmetry constraint lat togeth miur konopelchenko continu incarn matrix string equ miur cklund transform nat consequ und lat structure constrain system equ dimend gen hierarchy rel grad provid explicit repres hierarchy includ assocy algebr second hamilton structure term fre cur
lect not devot the equ assocy describ geometry modul spac topolog field the introduc lect wdvv equ frobeni manifold appendix polynom solv wdvv appendix symmetriy wdvv twist frobeni manifold appendix wdvv chazy equ affin connect curv project structure lect topolog conform field the modul lect spac isomonodrom deform frobeni manifold appendix geometry flat pencil met appendix wdvv painlev appendix branch solv equ isomonodrom deform braid group appendix monodrom group frobeni manifold appendix gen hypergeomet equ assocy frobeni manifold monodrom appendix determin superpot frobeni manifold lect frobeni structure spac orbit coxet group appendix extend complex crystallograph group twist frobeni manifold lect diff geometry hurwitz spac lect frobeni manifold integr hierarchy coupl topolog grav
describ vect valu conjug equ funct group cas compact simpl group affin group construct funct weight trac certain intertwin op repres compact group pet weyl theorem imply equ funct writ linear combin trac next comput rad part laplac op act conjug equ funct obtain compl tely integr quant system matrix coefficy spec cas coincid trigonomet calogero sutherland mos mult partic system affin group cas prov spac equ funct fix homog degr respect act cent group finit dimend span weight trac intertwin op spac coincid spac wess zumino wit conform block ellipt curv comput rad part second ord laplac op affin group act equ funct find certain parabol part diff op deg ellipt calogero sutherland mos hamilton cent charg tend min dual coxet numb crit level quant integr hamilton obtain rad part high sugawar op cent crit level
pap introduc tim matrix epsilon quant group transform conjug repres standard form abl comput explicit form import quant bicov diff calcul deform structure const deform transposit op lambd gaug cov condit defin gen deform kil form deform chern class quant group us rel gen deform kil form pres term deform homotop op abl comput deform chern simon condit furtherm deform cocyc hierarchy deform gaug cov lagrang deform yang mil equ der
first part pap giv denomin id simpl finit dimend sup algebra frak deg inv bilinear form giv also charact sup dimend formula finit dimend irreduc frak mod atyp second part pap giv denomin id affin superalgebra frak assocy frak spec id giv almost many formula numb repres integ sum squ sum triangul numb introduc not integr frak mod giv class irreduc integr highest weight frak mod
study dimend the defin observ rel knot link quant expect valu observ giv coefficy alexand conway polynom
chern simon gaug the compact semisimpl group analys perturb the point view gen form perturb sery expand wilson lin pres term casimir op gaug group expand num knot inv obtain knot inv turn finit typ vassiliev inv possess integr repres us known result jon homf kauffm akutsu wadat polynom inv knot inv comput typ prim knot cross result suggest knot inv norm integ valu
consid model grav coefficy einstein hilbert act imagin part equ introduc thet vacu structure gen expand whos effect convert expand sery altern sign presum borel sum show spec heat model phys behavio repres nonperturb sery term integr modul spac punct spheres sery rewrit un integr suit modul spac infinit punct spheres explicit real fri shenk quant grav conject expand term punct gen expand der us duisterma heckm theorem brief analys expand term punct spheres also multicrit model
high dimend analog dispersionless hierarchy introduc addit dimend phas spac vary dispersionless hierarchy hierarchy extr spat dimend compact ev dimend tor integr hierarchy ex infinit dimend addit symmetry ens und twist theoret structure nonlinear rieman hilbert problem analog funct whos logarithm giv funct fre energy prepot contest matrix model topolog conform field the construct infinit dimend symmetry extend funct extend symmetry lik dispersionless hierarchy obey anom commut rel
tod lat hierarchy assocy matrix form boson hierarchy provid framework drinfeld sokolov reduc scheme real hamilton act within second poisson bracket work fre cur abel second hamilton structure abl obtain un form reduc hierarchy interpol ordin hierarchy correspond op giv superdetermin grad mat diagon gaug describ bracket structure field cont particul provid explicit fre field repres assocy poisson bracket algebra gen famili nonlinear algebr discret cklund transform gen nat lat transl und tod lik hierarchy apply demonst equ matrix string model hierarchy
pres un approach thermodynam beth ansatz magnet chain field the includ finit siz zero temp calc lat model cas fre energy follow quadr solv singl linear integr equ nli system nli appear nest der nli vertex model twist bound condit chain extern magnet field sin gordon mass thir model period siz equ us light con approach nli solv it regim high chain model opposit conform regim lead behavy obtain clos form high correct der rieman hilbert form nli pres
consid spin model antiferomagnet regim us fre field real quant affin algebr level giv fre field real typ vertex op describ cre annihil phys partic model tak trac typ typ vertex op irreduc highest weight repres also der integr formul form fact model investig structure pol obtain residu formul form fact lat analog high spin extend smirnov formul mass integr quant field the result wel quant deform knizhnik zamolodchikov equ form fact show deep connect mathem structure integr lat model mass integr quant field the
pap pres noncommut the symmet funct bas not quas determin begin form the correspond cas symmet funct infinit numb independ vary allow endow result algebr hopf structure lead method comput desc algebra also giv un reinterpret numb class construct next study noncommut analog symmet polynom ar diff construct accord particul kind apply consid exampl polynom noncommut coefficy cent vary decompos produc linear fact root fact diff expand polynom thu accord wheth interest construct polynom giv root expand produc linear fact consid distinct spec form symmet funct third typ appear look noncommut gen apply rel not charact polynom matrix construct apply inst noncommut mat form gen univers envelop algebr
work construct cohomolog the control deform gen drinfel algebr task accompl three step first step construct mod cob complex adapt coassocy comultiply follow step involv high triv construct first construct ess combin defin diff grad algebr structure simpl chain complex associahedr second construct algebra nat definit diff grad algebra complex defin algebr der resolv us ex acyc associahedr defin cal homotop comod structure resolv gen drinfeld algebr turn allow defin desir cohomolog the term complex consist rough speak bimod bicomod map resolv mod cob resolv complex bigrad bicomplex gerstenhab schack the bialgebr deform compon cobound op defin construct ment apply show drinfel deform univers envelop algebr simpl algebr jump deform result pap annount pap drinfel algebr deform associahedr imrn duk math journ appear also preprint
study rel among vary form oscil algebr consid condit support hopf structure also pres gen algebr togeth correspond hopf structure multimod extend also consid
the repres clifford algebra extend employ divid algebr octon cayley numb particul quest ar assocy commut divid algebr answ octon repres clifford algebra lead not octon spin us giv octon repres respect orthogon group fin tri automorph shown exhibit manifest perm tim structure framework
pres polynom id imply gen eul rog ramanud id sid id interpret gen funct certain restrict partit prov id est graph correspond kind restrict partit
construct euclid hopf algebr dual lcross real subalgebr diff algebr quant euclid spac fact extend prevy real cit within introduc der gen transl funda hilbert spac repres turn highest weight typ rath simpl lat regul vert class on vect bas singlet zero spin irrep real norm funct going distribut limit rightarrow
typ simpl superalgebra study quant deform untw affin extend ident addit rel simpl gen extr ser rel nee impos prop defin uqgh pres gen techn der spect paramet depend mat quant affin superalgebra determin mat typ affin superalgebr vary repres thereby der solv spect depend yang baxt equ particul algebr possess paramet famy finit dimend irrep abl construct mat depend addit spect lik paramet provid gen fre ferm model
review multiparamet linear quant group real form dual algebr bicov diff calcul giv first part pap construct multiparamet linear inhomog quant group project equ quoty respect suit hopf algebr id bicov diff calcul explicit obtain project proc un singl structure quant plan coordin group matrix el allow deduc without effort diff calcul plan gen the illust exampl
prov sery id boson ferm repres certain virasoro charact id includ conject mad stony brook group spec cas method direct apply andrew extend bailey lemm rec obtain polynom id
invit lect intern congress mathem zuerich august extend vert review fre field real affin moody algebra apply
study dilogarithm id algebra analys asymptot theoret combin repres theoret point view prov dilogarithm id hypothet obtain us fiv term rel among coxet lewin loxton browkin on contain access lewin vary multivary funct equ giv level cas kunib nakanish dilogarithm conject prov addit result remaind term obtain connect dilogarithm id rog ramanud andrew gordon typ partit id asymptot behavy discuss result string funct level vacu repres affin algebr obtain connect dilogarithm id algebra the tors discuss rel cryst bas branch funct lambd lambd kostk foulk polynom lusztig analog weight multipl consid melz miln conject prov spec cas prov branch funct lambd lambd eq appropry limit kostk polynom cal thermodynam beth ansatz limit connect finit dimend part cryst bas robinson schensted knu correspond consid
dilut lat model regim univers class is model magnet field est direct ex structure dilut model regim express dimend config sum term ferm sum explicit involv root system thermodynam limit polynom id yield proof rog ramanud id rec conject kedem polynom id also apply regim obtain transform modul paramet cas find tim mbox structure prov rog ramanud id tim mbox typ fin crit limit giv intrigu express numb step path dynkin diagram tadpol term cart matrix find confirm structure dilut model found rec mean thermodynam beth ansatz
analog hurwitz zet funct introduc consid spect zet funct quant group analys aspect study eul maclaurin sum formul asymptot formula relev funct discuss
rigid framework cart calcul der in der funct form propos construct employ sem direct produc grad hopf algebra respect sup extend deform univers envelop algebr algebr funct quant group rel cart calcul follow consist condit approach prior bas leibn rul extery der might hent also interest rec work deform cart id der prov artic bas lect giv enrico ferm sum school quant group varenn jun
un system scheme constract diff opr real irreduc repres develop analog un scheme us constract diff op real irreduc repres explicit result giv
recal abstract the hopf algebr bicrossproduc doubl cross produc auth develop less wel known result quant doubl twist extend lorentz group
giv overview kind symmetry phys ex observ stat mad poss langu hopf algebra quant geometry propos auth feat planck scal phys rec work includ correspond result semiclass level poisson group level braid group braid geometry
structures quant braid spac typ defin matrix study includ minkowsk euclid spac addit braid group dual braid group vect covect prov first apply braid geometry mad
pap reemphas ubiquit rol modul spac algebra curv assocy algebr algebra topolog main result spac operad multiply homotop homotop grad poisson algebr singul cochain complex nat operad operad dec modul spac act nat rham complex omeg bullet hler manifold thereby yield gen typ homotop algebr structure omeg bullet reason operad nonsens phys bulletin board typ construct supersymmet sigm model construct gromov wit inv kontsevich vert
gen class linear diff finit diff op possess finit dimend inv subspac polynom bas giv main result op property must repres polynom el univers envelop algebr algebr diff diff op finit dimend repres dimend cas class giv algebra bold diff op bold bold finit diff op bold op real grassman vary equ tim matrix op bold bold op contain diff op par op class linear op possess infinit many finit dimend inv subspac bas polynom pres
list chapt introduc not prelimin part finit quant group matrix quant group quant plan quant envelop algebra root un part oscil repres oscil root un oscil root un part infinit quant group quant affin algebra quant affin algebra root un
system laplac casimir op orthogon symplect algebra construct op express term path graph rel mat form gen algebra property noncommut symmet funct assocy matrix decomposit sklyanin determin produc quas determin play main rol construct analog decomposit quant determin provid altern proof known construct algebr
rec scal funct dilut contract self avoid polym loop cylind rel painlev transcend us perturb the thermidynam beth ansatz num calc argu simil rel contract self avoid loop
diagon transf matrix inhomog vertex model vertex typ ant ferroelect regim intoduc typ vertex op spec cas model us diagon exchang model cit vertex op construct level vertex op fus proc describ boson ord clar partic structure estab isomorph cryst result simpl fig repres theoret ground stat deg
us techn develop fronsd galindo let math phy study hopf dual quant algebra hopf structure dual der correspond univers matrix embody suit mod expon rel rightarrow obtain
provid proof complet describ auth construct stat inv annount gen arbit artin semisimpl tortil categ also discuss rel inv gen brod surgery inv us techn develop cas sem simpl quoty princip root un robert brief discuss gen inv manifold equip dimend homolog class introduc yet robert subject sequel cit ref bibliograph pap
consid interpret class geometry conform field the construct orbifold discret tors exampl analys spacetim contain stringy reg class point view singul neith resolv blown model also giv particul simpl clear exampl mir symmetry
pap brief review rec result conceiv gen funct defin gen funct matrix el giv highest weight repres univers envelop algebr despit diff particul cas conv funct integr tod lat hierarchy gen funct also satisfy bilinear hirot lik equ deduc manip intertwin op main exampl consid detail cas quant group funct numb tak valu commut algebra funct quant group pap contain illust calc simplest cas algebr quant counterpart wel system funda repres
study cohomolog schwinger term ar second quant class observ belong restrict gen linear algebr prov pseudodiff op dimend typ schwinger term equ twist rad cocyc mod vert rad cocyc ar commut diff geometry process also show ordin rad cocyc pair pseudodiff op dimend writ phas spac integr star commut symbol project appropry asymptot compon
assocy standard mat introduc quant spheres project quant spac quant grassman manifold algebra shown homog quant spac standard quant group also quant principl bundl sens brzezinski majid com math phy
knizhnik zamolodchikov bernard equ twist conform block compact rieman surfac mark point writ explicit gen project structure term correl funct the twist system check modul spac equ provid flat connect spect paramet
construct class repres quadr matrix algebr giv reflect equ spect paramet term certain ordin diff op op turn paramet shift op hypergeomet funct limit cas class orthogon polynom rel fact method discuss
ex many four dimend integr the includ self dual gaug grav the extend supersymmet gen wel ful self dual sup yang mil equ review harmon spac form twist transform the yield method produc explicit connect met form us conceiv harmon spac analys clos rel quatern analys talk ogievetsky ursey mem conf istanb jun
rec techn harmon analys symmet spac suggest bas certain remark repres affin doubl affin heck algebra term dunkl demaz op instead group algebra class cas result among apply the rad part laplac op deform includ rel conceiv foury transform pres pap demonst techn work wel ev gen diff ellipt cas conject correspond moody algebra discuss construct gen rad zon laplac op connect diff ellipt ruijsena op gen turn olshanetsky perelomov diff ellipt op
gen definit propos rec linear connect met noncommut geometry shown within norm un linear connect quant plan met
octon produc giv octon flex found real divid algebra pattern flex investig
structure const minim conform the reconsid shown ratio structure const spin zero field diagon the evalu diagon the giv simpl express term compon eigenvect adjac matrix correspond dynkin diagram prov inspect lead car determin sign structure const appear form work subject also pres proof rely consid lat correl funct spec extend id comply the
pres system introduc geometry linear braid spac vert coordin braid stat describ matrix start point survey auth braid approach deform braid differenty expon gauss integr form bas ingredy deform phys cov braid approach includ nat euclid minkowsk spac matrix form
giv explicit describ vect bundl conform block ellipt curv mark point subbundl vect bundl weyl group inv vect valu thet funct cart subalgebr giv part conject charact subbundl term certain van condit affin hyperpl cas explicit calc poss confirm conject fri shenk flat connect calc shown horizont sect solv bernard gen knizhnik zamolodchikov equ
propos nonstandard approach solv app incompat coalgebr structure inhomog quant group nat complex conjug work sketch gen ide develop method detail model lat deform hopf algebr transl dil show hilbert spac repres lat tens produc funda on phys correspond construct composit system many fre distinct partic term bas partic on spectr tot moment turn partic moment form
conject polynom id imply rog ramanud typ id branch funct assocy coset otim mbox support conject est correct behavio level rank dual show rog ramanud id expect asymptot term dilogarithm id poss gen arbit coset also discuss brief
ful dual kapp poinc algebr kapp poinc group prov
giv simpl algebr gggg consid orbit gggg matrix typ possess poisson pencil gen kirillov kost souriau bracket cal matrix bracket cal algebr quant lat bracket quant orbit matrix typ describ orbit typ explicit construct quant whol poisson pencil orbit simil not deform bracket braid coadjoint vect field tang vect field discuss wel
pap construct explicit nat geomet point view fock spac repres contragrady verm mod quant envelop algebra ord start gauss decomposit quant group introduc diff op correspond deform flag manifold asuum left comod quant group project right act quant envelop algebr quant group fin express repres el quant envelop algebr correspond left inv vect field quant group first ord diff op deform flag manifold
adl shiot moerbek obtain tod lat hierarchy formul transl act vertex op funct act vertex op pseudo diff op wav funct rel addit symmetry tod lat hierarchyto infty respect infty tim infty algebr sym pap gen result compon hierarchy vertex op gen algebr infty matrix vert infty sint tod lat hierarchy equ compon hierarchy result pap uncov particul cas much rich structure obtain adl shiot moerbek
every partit cdot assocy vertex op real algebra infty us construct obtain reduc compon hierarchy reduc rel partit obtain matrix typ equ show follow constraint funct equ funct ldot reduc hierarchy satisfy string equ satisfy vacu constraint infty algebr talk giv intern conf mathem phys string the quant grav alusht jun
bicov first ord diff calcul quant group determin distinct paramet famy calcul term suit bas form commut rel express help matrix calcul induc bicov diff calcul real form gen deform paramet calcul class limit bicov calcul ordin calcul obtain deform involv cart kil met
construct approach propos prevy pap allow pres pap construct gen deform paramet finit dimend repres quant superalgebr finit dimend mod construct eith irreduc indecompos mod indecompos condit hold ex inv maxim submod fact repres fact mod irreduc cal nontyp pap indecompos repres nontyp finit dimend repres quant superalgebr consid class mod structures an matrix el nontyp repres writ explicit
rec develop understand supersymmet yang mil the four dimend suggest point view donaldson the four manifold instead defin four manifold inv count instanton defin equ four manifold inv count solv linear equ abel gaug group dual equ gaug group dual maxim tor viewpoint suggest many result donaldson inv
kapp deform poinc superalgebr writ hopf superalgebr form transform bas class lorentz subalgebr gen show bas kapp deform poinc superalgebr writ grad bicrossproduc show kapp deform superalgebr act cov kapp deform chir superspac
cyc quant dilogarithm interpret cyc symbol weyl algebr consid borel subalgebr us mod symbol inv triang link triang manifold construct app amby isotop inv link
study hierarchy prov ex adl shiot moerbek formul formul rel act infty algebr funct act addit symmetry wav funct
certain situ giv geomet background quasiclass calc bas explicit connect quant mech collaps coh stat coadjoint orbit class op
us fronsd galindo formul expon map quant algebr quant group show dimend repres obtain exponenty wel known dimend repres frac ldot frac correspond defin dimend matrix ear result finit dimend repres obtain repres backslash backslash also consid structure clebsch gord matrix study clebsch gord coefficy appl reduc direct produc repres quant group
describ infinit famy mult paramet funct equ rog dilogarithm gen abel eul formula suggest thermodynam beth ansatz approach renorm group flow integr rel quant field the known rul cent charg crit fix point obtain spec cas conject simil funct id construct rat integr quant field the fact matrix support extend num check
study arithmet property mir map quant yukaw coupl paramet deform calab manifold first schwarzian diff equ der prevy charact mir cas algebra surfac solv equ term funct der explicit modul rel prov mir map algebra gen zero funct field lead uniform proof mir map integr foury coefficy regard map rieman map prov gen zero funct virtu conway norton conject prov borcherd us frenkel lepowsky meurm moonshin mod find map act reciproc thompson sery certain conjug class griess fisch group also giv immedy consequ second proof mir map integr thu conject surpr connect mir map thompson sery threefolds construct form nonlinear quant coupl reduc mir hypothes integr assum der congru foury coefficy quint deduc least degr instanton numb divis fact first conject clem
typ quant superalgebra known admit triv paramet famy inequ finit dimend irrep ev gen apply rec develop techn construct solv quant yang baxt equ assocy paramet famy irrep thu obtain mat depend spect paramet addit continu paramet extr paramet ent yang baxt equ simil spect paramet addit form
show reason wel behav tqfts must contain certain algebra structures find hopf categ trialgebra
contract deform real construct deform dimend poinc algebr algebr funct group diff structure also shown hopf algebr triangul univers matrix also construct explicit find deform univers envelop algebr giv deform mass shel lorentz transform
first sery pap study combin subtract bas charact standard mod affin algebra wel vary subspac coset spac mod part consid certain standard mod affin algebr posit integr level construct bas princip subspac introduc study rec feigin stoyanovsky bas giv term partit col charg assign part partit part col charg comply certain diff condit part repres foury coefficy vertex op interpret quas partic enjoy partic stat interact rel cart matrix particul cas vacu mod charact formul assocy bas annount combin charact propos whol standard vacu mod level
investig inhomog quant group built quant group transl correspond commut rel contain inhomog term certain condit satisfy study quant poinc group prov construct correct siz find mat analog minkowsk spac
us gen the quant poinc group without dil describ investig describ contain num paramet satisfy certain polynom equ cas solv giv class quant poinc group correspond exact quant minkowsk spac poinc sery object class cas also class poss mat funda repres group
der larg asymptot surgery formul wit inv gen seifert manifold contribut connect compon modul spac flat connect ident contribut irreduc connect pres residu form form simil us szen jeffrey kirw simil allow express contribut irreduc connect term intersect numb modul spac
bialgebr contract introduc class deg cobound bialgebr structure impl pseudo orthogon algebra start correspond allow introduc bialgebr contract lead bialgebra quas orthogon algebra construct explicit giv cas bialgebr contract study pap defin hopf algebr contract drinfel jimbo deform explicit us gen semisimpl quant algebra cas euclid poinc galil algebra
show knot inv obtain chern simon the bas arbit abel gaug group distinct isotop inequ mut knot link attempt distinct knot link study murakam symmet vert strand composit braid saly feat the composit braid pres repres gen braid obtain exploit property hilbert spac assocy correl wess zumino conform field the composit inv knot giv el chern simon inv assocy irreduc repres produc repres allow fus rul correspond wess zumino conform field the plac individ strands composit braid hand composit inv link giv weight el multicolo chern simon inv mut link distinct composit inv mut knot shar property result though develop detail within framework chern simon the valid abel gaug group
pres almost complet list norm form class mat poinc group
rec group list set weight weight project spac admit transvers calab hypersurfac not correspond calab manifold form mir symmet sint manifold hodg numb whos mir occ list mean batyrev construct check manifold indee mir miss mir construct hypersurfac tor vary show many manifold interpret transvers hypersurfac weight hypersurfac van point origin fal outsid us rang landau ginzburg the nevertheless batyrev proc provid mak sens the
extend braid rel multipl braid tens produc algebra us quant nonultraloc model pres yang baxt typ consist condit wel condit ex multipl coproduc monodrom matrix us construct commut subalgebr giv homog loc algebra introduc simpl yang baxt typ condit shown yang baxt typ equ multipl coproduc condit homog loc ord algebra solv
diff geometry hopf algebr construct us bas axiom hopf algebra noncommut diff geometry spac gen der hopf algebr funct pres smash produc us defin discuss quant algebra property cart calcul extery der der in der found univers gen diff calcul arbit hopf algebr restrict quasitriangul cas us num matrix form af structures quant group determin
gaug linear sigm model introduc wit calc instanton expand correl funct topolog sigm model target spac tor vary calab hypersurfac subset linear model instanton modul spac rel simpl object correl explicit comput moreov instanton sum lead explicit solv kind model cas smoo result reproduc clar algebra solv model batyrev addit find algebra rel determin solv term fin propos mod linear model comput instanton expand limit point modul spac smoo cas lead second algebra solv model describ prov conject mir symmetry includ prevy conject monom div mir aspinwal green morrison
review loc techn funct integr rec us perform calc gain insight structure certain topolog field the dimend gaug the funct integr counterpart matha quil form duisterma heckm theorem weyl integr formul respect cas first introduc necess mathem background eul class vect bundl equ cohomolog topolog group describ finit dimend integr formula discuss apply path integr giv overview relev lit apply deal includ supersymmet quant mech cohomolog field the phas spac path integr dimend yang mil the
study cohomolog ar brst form gaug wznw model stat gaug the project ungaug mean brst condit der gen simpl group arbit level condit cohomolog triv show introduc smal perturb jantz highest weight repres stat cohomolog singlet pair ar unphys stat kugo ojim quartet perturb zero en ident construct stat cohomolog ghost numb occ un determin repres algebra involv construct giv term cur mod rely explicit form highest weight nul stat giv malikov feig fuch
hamilton jacob form gen dimend field the accord lep canon framework apply sev rel real scal field massless mass klein gordon sinh sin gordon liouvil the rel eul lagrang hamilton jacob equ discuss dedond weyl correspond wav front calc cara eod form unlik mech impos certain integr condit veloc field guar transvers rel espec dynam equ hamilton jacob wav front famy extrem embed therein acklund transform play cruc rol solv result system coupl nonlinear pde
find volum form modul spac doubl punct rieman surfac whos integr satisfy painlev recurs rel gen expand spec heat grav allow express asymptot expand spec heat integr infinit dimend modul spac spirit fri shenk approach outlin conject der recurs rel us duisterma heckm theorem
study first ord form pur four dimend yang mil the thet term describ green funct assocy elect magnet flux op hooft mean gaug inv loc op green funct rel wit inv four manifold
study path integr holomorph yang mil the compact hler surfac bas result examin correl funct topolog yang mil the correspond donaldson inv wel transit formula
rec work sing dougla gopakum gross apply result voiculescu commut prob the construct mast field larg matrix field the suggest not consid interrel mast field quant group defin mast field algebr observ isomorph algebr funct quant group mast field becom cent el quant group hopf algebr quant haar meas giv wign semicirc distribut mast field coh stat becom coh stat mast field the
show construct exact fact mat dimend quant field the whos symmetry charg gen quant affin algebr quant affin tod the exampl the tak account lorentz spin symmetry charg determin grad quant affin algebra giv mat rigid pol structure depend kind quant dual coxet numb theref also determin quant mass ratio the exampl explicit construct mat symmetry
show spin gen ellipt calogero mos system ellipt extend gaudin model cousin tre deg hitchin system apply construct integr mot angl act vary quant system discuss explicit formula op high gen surfac obtain shottky paramet construct mot conform field the quant counterpart tre deg crit level knizhnik zamolodchikov bernard equ
simpl connect univers matrix spin dem requir form produc act hilbert spac gen quant group symmetry forward giv explicit op real produc act cov op allow der quant group cov fus braid mat although typ gen depend upon worldsheet vary obey cent extend real cal fix point commut rel explain show link algebr field transform produc gen much weak prevy thought cent charg extend algebr includ liouvil zero mod moment nont rel virasoro descend un also show approach us der hopf algebr structure extend quant group symmetry odot qhat rel pres screening charg grav
study twist supersymmet yang mil the coupl hypermultiplet tqcd suggest famy tqcd serv pow tool study quant field theoret property und phys the
study topolog obstruct stringy nat concern deform target spac linear target spac singul smooth away accord conv rul geometry study assocy conform field the see deform poss without discontinu chang correl funct obstruct appear com tors homolog target spac seen deform the irrelev op discuss link phenomenon orbifold discret tors study vaf wit
tim matrix vert hierarchy rec tre reduc system ar drinfeld sokolov typ hamilton symmetry reduc apply poisson submanifold dual algebr wideh otim complex lambd lambd sery extend matrix gelfand dickey system der mean gen drinfeld sokolov reduc defin algebr wideh otim complex lambd lambd us nat embed subset posit integ hierarchy obtain admit describ term tim matrix pseudo diff op compr typ posit part triv neg part system investig prevy cas constrain system pap prevy result consid extend system study pres bas drinfeld sokolov approach adv lead loc poisson bracket mak clear conform algebr structures rel typ hierarchy discret reduc mod vert extend hierarchy also discuss
pres abel gen wit monopol equ analys assocy modul problem regard gen donaldson the modul spac solv monopol ahl manifold discuss also construct us matha quil form topolog quant field the correspond modul problem the involv coupl topolog yang mil the topolog mat four dimend
quant deform oscil algebr imply phas op study view point index theorem us explicit matrix repres posit deform paramet thet ir thet obtain index condit dag allow hermit phas op expon varph expon varph dag thet rat thet form obtain singul situ infty dag infty allow hermit phas op expon expon dag wel hermit expon varph expon varph dag imply interpret quant deform discuss also show overcom problem neg norm thet
matrix sin gordon the matrix gen wel known sin gordon the study particul gen field tak valu describ integr deform conform field the correspond coset tim vary class aspect matrix sin gordon the address find exact solv soliton breath gen sin gordon the intern degr freedom apply zakharov shab dress method explain phys property infinit cur conserv law cklund transform the obtain zero curv form equ mot cklund transform also der exact solv wel nonlinear superposit principl mak bianch permut theorem
pap discuss topolog the dimend observ assocy ordin knot link dimend knot dimend vacu expect valu observ giv wid rang inv consid main dimend cas inv includ alexand polynom homf polynom kontsevich integr
pres pap obtain integr gen tod system gen flat connect form tak valu high grad subspac simpl algebr construct gen solv think system describ field mat typ coupl standard tod system spec interest nonabel tod the lat involv black hol target spac met also giv der conform system bas hamilton reduc wznw model discuss rel abel nonabel system gen gaug transform map first grad describ second lat involv grad larg
prov eq stat phas conform un superselect sect finit index conform quant field the relev point describ symmetry construct glob conjug charg
study tens produc funda repres yang show funda quoty tens produc giv dorey rul
mult vert overlap id satisfy exploit understand property hyperbol moody algebra particul sint algebr embed larg algebr phys stat assocy complet compact subcrit boson string principl determin root spac analys posit norm phys stat decoupl string vertex consequ algebr phys stat decompos direct hyperbol algebr spac decoupl stat spac contain transvers longitudin stat longitudin decoupl hold gen also valid uncompact strings poss consequ liouvil the id decoupl stat simply amount find zero certain decoupl polynom cas transvers decoupl cruc depend spec property root lat explicit demonst triv root spac vert compact string contain complet inform decoupl property hyperbol algebr encod view integ grad hyperbol algebra level algebra interpret interact strings mov respect group manifold assocy und finit dimend algebra
bas dispersionless the giv comprehend study topolog landau ginzburg the charact rat pot writ hierarchy gen form find hierarchy nat includ dispersionless continu limit tod hierarchy gen finit numb prim sev flat solv topolog the obtain form ident discuss dubrovin explicit construct gravit descend prim field giv residu formul point funct field show point funct satisfy topolog recurs rel string equ obtain gen hodograph solv hierarchy show gravit effect constitut equ point funct renorm coupl const smal phas spac
bas class the envelop form renorm group method glob analys rec propos goldenfeld clar equ improv thing
gaug the known possess elect magnet dual symmetry invert coupl const extend act pap dual study gen four manifold shown partit funct modul inv funct transform modul form result play ess rol determin energy interact ar supersymmet yang mil the form four manifold determin interact giv test solv model would ent comput donaldson inv four manifold certain aspect abel dual relev mat depend donaldson inv second stieffel whitney class also analys
pres altern topolog twist yang mil path integr domin instanton flat connect complex gaug group the nont compact ory four manifold nonposit eul numb necess simply connect manifold find singl topolog inv analog casson inv three manifold
dispersionless diff id shown equ kernel expand provid univers algebra charact solv dispersionless hirot equ calc bas dat act also ind
modul depend superst compact bas calab hypersurfac weight project spac investig ferm typ polynom constraint correspond landau ginzburg orbifold whos pot typ singul consid gen arbit quas homog singul mir symmetry der depend model complex ahl modul check expand topolog correl funct explicit gen zero gen instanton calc import apply giv exampl algebra twist deform map algebra on hent allow study ful modul spac also study modul spac nest thu en singul transit the anoth follow rec work green morrison strominger show correspond black hol condens typ string the compact calab manifold
transl inv symmet polynom coordin body problem id partic propos shown coordin calogero sutherland body hamilton appropry gaug transform pres quadr polynom gen algebr finit dimend deg repres exact solv model follow ex infinit flag repres spac preserv hamilton connect jack polynom discuss
automorph dynkin diagram symmet moody algebr induc automorph algebr map highest weight mod larg class dynkin diagram automorph describ vary aspect map term anoth moody algebr orbit algebr particul gen funct trac mod twin charact eq charact orbit algebr orbit algebra twin charact constitut cruc step toward solv fix point resolv problem conform field the
analys beth ansatz execut wid class finit dimend mod label skew young diagram gen contain frag correspond spin repres transf matrix spectr relev vertex model est numb formula analog class on jacob trud giambell schur funct yield ful solv prevy propos funct rel system tod equ
fract supersymmetry denot gen supersymmetry construct us singl real gen grassman vary thet thet thet arbit integ explicit formul giv cas gen transform leav the inv shown transform possess interest group property shown also gen der ent the geomet interpret gen left right transform fract supersymmetry group car at paid techn import issu includ differenty ar result peculi nat quant thet
consid system ar thermodynam beth ansatz prov period solv system conject zamolodchikov dilogarithm id conject gliozz tateo
not modul cov review reconstruct poinc group extend dimend cas rel symmetry spin stat theorem describ
prov polynom id superconform model gen extend known ferm bos charact id proof us trinom coefficy andrew baxt boson sid rec introduc gen method produc recurs rel sery ferm sid polynom demonst dual rel rightarrow also introduc gen wit index express term rog fals thet funct
geomet feat landau ginzburg orbifold pres model typ typ superpot show correspond stat charg integr point dual polyhedr us tool construct mir manifold rely tor geometry stat shown correspond form com blow process term id check monom div mir landau ginzburg orbifold propos auth equ mir calab manifold obtain mathem
pres explicit express topolog inv assocy monopol funda repres spin four manifold comput inv bas analys correspond topolog quant field the turn express term seiberg wit inv analys rec exact result modul spac vacu untw supersymmet counterpart topolog quant field the consid wel elect magnet dual supersymmet gaug the
spin gen rel calogero sutherland model construct us affin heck algebr shown possess quant affin symmetry uqglt spin less model exact diagon mean macdonald symmet polynom dynam dens dens correl funct wel partic green funct evalu exact also investig finit siz scal model show energy behavy describ gauss the result ind excit obey fract exclud stat exhibit tomonag lut liquid behavy wel
study sev aspect canon quant supergrav term asthek vary cast the term connect introduc loop repres solv spac simil loop repres ordin grav main diff form mandelstam id phys stat gen giv knot inv compat mandelstam id explicit solv quant constraint equ connect chern simon form coincid exact dubrovnik vert kauffm polynom provid first tim poss find explicit analys express coefficy knot polynom
propos solv problem compat bos ferm stat symmetry transform impl compact quant group drinfel typ unit transform conjug mult partic symmetry post obtain twist real symmet group
poisson bracket algebr correspond second hamilton structure larg class gen mkdv integr hierarchy car analys algebra known conform property rel algebra prevy investig particul cas class equ consid includ pract gen drinfel sokolov hierarchy construct lit particul rec shown includ matrix gen gelfand dickey constrain hierarchy theref result provid un describ rel hamilton structure soliton equ algebra compr almost result form obtain auth main result pap explicit gen equ show second poisson bracket algebr deform dirac bracket algebr correspond algebra obtain hamilton reduc
reduc symmetry linear system self dual yang mil equ four dimend repres conjug class subgroup connect part id correspond euclid group carry subgroup lead system diff equ nonequ condit zero curv without paramet system uncoupl first ord linear consid pair mod form nahm equ wel system part diff equ three dimend writ
mot rec find wiegman zabrodin faddeev kashaev concern appear quant symmetry problem bloch electron dimend magnet lat introduc mod tight bind azbel hofstadt hamilton spec spin eul consid class analog eigenvalu problem propos model coh stat repres describ equ thu complet solv observ striking simil shap spectr model vary valu spin
study ex monopol bound stat sat bound supersymmet yang mil the describ ex bound stat rel topolog index bundl modul spac solv us index theorem prov ex certain stat predict seiberg wit bas study vacu structure yang mil the
study path integr twist supersymmet yang mil the coupl hypermultiplet bar mass explicit comput topolog correl funct the compact ory simply connect simpl typ rieman manifold corol determin topolog correl funct the without bar mass the without coupl hypermultiplet includ concret field theoret proof rel donaldson seiberg wit inv
giv describ discret modul spac mcdisc introduc cit term discret rham cohomolog modul spac gen funct intersect ind cohomolog class found class highest degr coincid on continu modul spac show matrix model techn kontsevich matrix model gen funct continu cas matrix model pot alph bigl bigr latest cas effect delign mumford reduc becom relev stratification proc ord express integr op spac mdisc term intersect ind calc compact spac mcdisc find solv constraint equ partit funct matrix model express tim appear depend ev tim cdot logarithm partit funct kontsevich model quadr diff op
consid class dimend tod equ discret spac tim ar funct rel commut famy transf mat solv lat model assocy class simpl algebr pres solv term pfaff determin view yang analog class jacob trud formul schur funct
form constrain hierarchy denot affin wideh matrix integr hierarchy gen drinfeld sokolov hierarchy us algebra approach includ grad structure gen drinfeld sokolov hierarchy abl find sev univers result valid hierarchy particul method yield clos express second bracket obtain dirac reduc untw affin moody cur algebr explicit exampl giv cas wideh clos express gen recurs op also obtain show isospect flow charact group accord semisimpl regul el cont cent kernel
fix point resolv problem solv diagon coset the prim field fix point resolv describ submod branch spac obtain eigenspac automorph impl field id comput charact modul matrix orbit algebra twin charact introduc prevy pap charact prim field express term branch funct twin charact allow express modul matrix mat orbit algebra assocy id group result extend larg class gen diagon coset
noncrit string model grav coupl fre scal field pap discuss brst quant direct analog virasoro string particul calc phys spectr problem brst cohomolog correspond op cohomolog form algebr model algebr polyd commut ring vary quadr constraint equ algebr polynom polyvect field bas affin spac pap attempt pres complet sum progress mad study
shown mccall hahn the self induc transp coh opt puls prop ident complex sin gordon the sharp lin limit reform the term deform gaug wess zumino wit sigm model address vary aspect self induc transp
investig analog the the turn surpr finit property infinit dimend cur algebr symmetry correl funct determin symmetry defin the system process quant modul spac holomorph vect bundl algebra surfac outlin defin vertex op the fin defin four dimend conform block pres analog verlind formul
gen drinfeld sokolov construct system review cas arbit affin algebr pay particul at hamilton aspect algebra extend known result wel interpret construct also pres
whilst many solv found quant yang baxt equ qyb few known solv avail high dimend gen zamolodchikov tetrahedron equ frenkel moor simplex equ pap pres famy solv help understand high dimend gen qyb
complet energy spectr obtain quant mech problem dimend eq mass partic interact pot alph sqrt shown scat config charact init moment goe fin config charact un fin moment
coulomb branch supersymmet gaug the four dimend describ gen integr hamilton system holomorph sens nat construct system com dimend gaug the spect curv start point view propos integr system relev gaug the hypermultiplet adjoint repres off much evid correct model dual group cent el act charg conjug permut hig confin obl confin phas expect fash also study exot phas
lat regul op nonultraloc mod korteweg vri mkdv equ propos quant level bas op satisfy deform braid algebr find assocy quant mat exact integr model prov braid quant yang baxt equ suit gen equ nonultraloc model us algebra beth ansatz eigenvalu problem quant mkdv model exact solv connect spin chain est facilit investig correspond conform property
smal correct uncertainty rel effect ultraviolet infr discuss context string the quant grav correct lead smal finit minim uncertainty posit moment meas shown effect could indee provid nat cutoff quant field the correspond und quant theoret framework includ smal noncommut geomet correct canon commut rel ord study ful imply conceiv loc cruc find phys stat maxim loc stat property calc cas minim uncertainty posit extend tre though stil dimend gen situ minim uncertainty posit moment
sev phys system id partic dimend isotrop oscil kepl system curv spac mathem structures quadr algebr finit algebr shown poss structure gen deform algebr repres the known furtherm gen deform paraferm oscil ident algebr sev phys system isotrop oscil kepl system curv spac foka lagerstrom smorodinsky winternitz holt pot mathem construct gen deform algebr finit algebra fact holt pot charact symmetry obtain produc
giv explicit express multinom gen binom andrew baxt trinom show config sum gen rso model regim study dat express term multinom gen work config sum stat mech model express term binom trinom coefficy rso config sum yield branch funct wideh tim wideh wideh coset model repres term multinom giv roch carid lik formula wherea repres dat giv doubl repres branch funct
clar rol play stat calc threshold correct heterot string compact evalu correct class compact show sum logarithm funct posit root gen moody algebra moreov certain limit formula suggest reform heterot string term gaug the bas hyperbol algebra defin gen moody superalgebr assocy stat fin discuss rel result string dual
mast field larg limit matrix model gaug the construct mast field satisfy standard equ rel field the field quant accord rul defin mast field yang feldm equ fre field quant boltzman fock spac mast field gaug the tak valu finit dimend algebr howev abel gaug symmetry construct mast field ess work minkowsk spac tim wightm correl funct brst quant mast field gaug the loop equ consid
calc numb curv calab manifold instanton expand alway agr would expect naiv explain account continu famy instanton deform the excess intersect the ess rol play deg instanton also explain pap slight expand vert auth talk jun triest conf dual mir symmetry
giv real quant affin algebr term anyon defin dimend lat deform paramet rel stat paramet anyon limit deform paramet going recov feingold frenkel ferm construct undeform affin algebr
brief introduc topolog quant field the wel describ rec progress mad field pres cont main connect chern simon gaug the vassiliev inv donaldson the gen seiberg wit inv emphas mad us rel obtain explicit express topolog inv univers structure und system
mod property spect mix bound condit massless spin half field der bal correspond funct determin heat kernel coefficy pres lat polynom
consid entrop rel entrop field the est relev monoton property respect coupl rel entrop field the hierarchy renorm group fix point rank fix point lowest rel entrop assign highest multicrit point argu consequ gen theorem wilson flow induc increas entrop propos rel entrop nat quant increas fix point anoth dimend
giv atiyah patod sing index the construct bundl ferm fock spac paramet vect pot spac dimend prov lead simpl man known schwinger term faddeev mickelsson cocyc gaug group act rel construct bundl gerb approach discuss rec carey murray includ explicit comput dixmy douady class adv method apply whenev form theorem hand cas ferm extern gravit field
period three form calab manifold found solv picard fuch equ howev tor variet method lead gen hypergeomet system equ solv period extend equ der symmetry consid sem period solv extend system obtain integr three form chain chain us construct cyc integr giv period simpl exampl abl obtain complet solv extend system also conject certain mod method gen ful spac solv gen
not high bundl gerb introduc giv geomet real high degr integr cohomolog certain manifold consid exampl us infinit dimend spac ar gaug the
pres explicit construct cent extend group compact manifold group complex curv obtain simpl construct extend picard vary construct easy adapt extend gaug group automorph nont vect bundl
larg asymptot perturb sery integr form smoo form smoo funct manifold inv act symmetry group comput us stat phas approxim perturb sery express integr form spac crit point act pap overview form feynm rul comput form proof perturb sery obtain independ cho met nee defin also overview definit adapt context dimend chern simon quant field the infinit dimend result construct diff inv depend clos ory manifold togeth cho smoo compon modul spac flat connect compact structure group mak pap access warm triv exampl giv outlin proof obtain inv chern simon cas ful detail appear elsewh
show modul spac calab manifold real hypersurfac describ transvers polynom four dimend weight project spac connect achiev exploit techn tor geometry construct batyrev rel calab manifold reflex polyhedr tak togeth prevy known fact modul spac cicy connect moreov connect modul spac pres class calab manifold sint quint threefold cicy hypersurfac weight strongly suggest modul spac simply connect calab manifold connect interest singul calab manifold correspond point modul spac meet oft pres class singul conifold connect modul spac cicy
construct conserv law chir model gen sigm model dimend spac tim continu us diff form extend also compr correspond discret vert achiev deform ordin diff calcul particul nonlinear tod lat result linear continu wav equ method apply sev exampl also construct pair acklund transform class model consid work
giv simpl simply lac complex algebr correspond group bfnp subalgebr gen posit root pap construct algebr whos und grad commut algebr giv cohomolog respect bfnp algebr regul funct valu mywedg bfnp backslash conject describ algebr phys brst inv op noncrit string conject ver explicit known cas sltw virasoro string slth string
legendr transform gen origin found supersymmet sigm model techn us giv construct hyperkahl met giv twist spac interpret gen legendr transform construct atiyah hitchin met modul spac monopol us detail exampl
gen deform oscil schemes introduc un framework vary deform oscil prov equ un repres lead correspond deform oscil supersymmet quant mech susy scheme addit sev phys system id partic dimend isotrop oscil kepl system curv spac mathem structures quadr algebr finit algebr shown possess structure gen deform algebr repres the known furtherm gen deform paraferm oscil ident algebr sev phys system isotrop oscil kepl system curv spac foka lagerstrom smorodinsky winternitz holt pot mathem construct gen deform algebr finit algebra fact holt pot charact symmetry obtain produc
investig high grad integr gen affin tod system extr field assocy zero grad gen obey field equ dirac typ regard mat field model possess soliton config interpret partic the foot assocy funda field spec subclass model remark possess noeth cur spec gaug fix conform symmetry proport topolog cur lead confin mat field insid soliton regard dimend model model also relev study electron self loc quas dimend electron phonon system
describ count stat typ strings rel supersymmet cyc gen numb rat curv doubl point gen funct numb curv left mov partit funct boson string
instanton solv forc burg equ found solv describ expon tail prob distribut funct veloc diff reg shock wav abs result agr found rec polyakov us op produc conject conject tru asymptot wyld funct integr exact ord perturb expand around instanton solv explicit check first ord also gen solv arbit dimend burg equ result found angul depend veloc diff
examin quant energy level rectangul billiard pointlik scat dimend vary loc strength scat system find diabol deg among vary level assocy berry phas illust ex loc wav funct point dimend ev ground stat shown display sign revers mech circumv sturm liouvil theorem
prov mir conject calab manifold equip holomorph symplect form manifold also known complex manifold hyperkaehl typ obtain complex manifold hyperkaehl typ mir dual mir conject stat follow kontsevich talk equ certain algebra structures rel vary hodg structures comput canon flat coordin modul spac calab manifold hyperkaehl typ introduc mir symmetry bershadsky cecott oogur vaf
reflex polyhedr encod combin dat mir pair calab hypersurfac tor vary investig geomet structures circumscrib polytop minim numb facet inscrib polytop minim numb vert object constrain reflex pair polyhedr intery extery describ term certain neg integr mat maj tool class mat ex pair weight system ind rel weight project spac corn ston algorithm construct dual pair reflex polyhedr expect efficy enough enum class dimend relev cas calab compact string the
form triv composit class gaug structures giv par gaug structures common bas spac synthes obtain daught structure funda model bas pair rel connect tak valu produc spac correspond algebra curv cov extery der assocy structural id contribut gaug group vary induc structures class whos composit giv triv mean possess irreduc nat pur irreduc piec particul gen complet sup spac ghost attend sup brst vary law pur geomet origin
gen solv grad contract equ otim grad real compact simpl algebr pres explicit turn depend independ real paramet structure gen grad contract display dimend cas kinem algebra shown appear straightforwardly geomet phys mean contract paramet curv also analys particul kinem algebra curv direct link geomet property poss homog spac tim
field the loc transform belong quant group defin class spacetim gaug pot belong quant algebr gaug transform defin pot lead appropry quant group transform field strengths cov der defin el mean adjoint act guar triv deform gaug inv commut rel ident
pres diagon solv reflect equ ellipt solv star triangl rel model consid rel affin algebra recov known diagon solv assocy algebra find solv rel ellipt regim furtherm solv reflect equ follow assocy vertex model trigonomet limit
show enh gaug symmetry typ string the compact calab threefold ar singul geometry target spac target spac typ string acquir gen curv singul find gaug the adjoint hypermultiplet appear singul massless stat correspond soliton wrap collaps cyc dynam describ twist supersymmet gaug the tim reproduc result analys dual manifold check predict mad model nat hig branch monodrom period integr asymptot loop topolog amplitud agr geomet comput exampl find singul occ strong coupl heterot dual propos kachru vaf
start deform quant star produc quant problem nambu mech investig consid imposs push analog field quant solv quant problem pres novel approach zarisk quant field observ funct cas polynom quant bas fact polynom sev real vary quant infinit dimend algebr field gen polynom defin deform algebr abel assocy distribut proc adapt der nee nambu bracket ens valid funda id nambu mech also quant level construct fact gen particul cas consid util quit gen defin id much gen star produc
addit isospect symmetry form constrain kadomtsev petviashvil integr hierarchy problem compat addit symmetry und constraint solv explicit virasoro part addit symmetry appropry mod standard addit symmetry flow gen unconstrain hierarchy also discuss spec cas trunt hierarchy obtain darboux cklund orbit init pur diff op lat giv ris tod lat lik structures relev discret mult matrix model construct est condit commut addit symmetry flow discret darboux cklund transform hierarchy lead der string equ constraint matrix model
us boson techn der integr repres mult point loc hight prob andrew baxt forrest model regim argu dynam symmetry model provid deform virasoro algebr
continu study quant field the noncommut geomet spac focus class noncommut geometry imply ultraviolet infr mod form nonzero minim uncertainty posit moment cas ultraviolet mod uncertainty rel appear string the quant grav cov exampl euclid the study detail exampl show ultraviolet infr regul graph
reform hamilton approach lat gaug the class level gaug group canon instead poisson group quant level get promot quant group gaug symmetry the depend paramet deform paramet lambd lat spac show system kogut susskind recov lambd rightarrow recov continu limit lambd thu poss paramet regul
demonst certain gaug mat rat hyperbol ruijsena schneider model quadr matrix poisson bracket exact quadr linear matrix poisson bracket calogero mos model phenomenon explain geomet der equ arbit flow hierarchy turn govern dynam op
expect valu wilson loop chern simon the knot inv skein rel der vary way includ vary method smal deform loop mad chang evalu lat method allow obtain approxim express skein rel pres gen ide allow comput exact form skein rel moreov requir gen result knot inv intersect knot link man consist mandelstam id satisfy wilson loop allow first tim der ful express knot inv suit candid quant stat grav supergrav loop repres approach lead sev insight intersect knot the particul rol plan intersect intersect kink
first exhibit commut cas simpl algebra rel algebr funct manifold infinitesim leng el unit repres correspond rieman met spin structure dirac prop dirac op extend simpl rel commut cas us tomit involv writ spect act trac funct leng el planck unit apply commut geometry standard model shown joint work chamseddin giv lagrang coupl grav intern fluctu commut geometry triv commut cas yield ful boson sect correct quant numb slight commut cas group loc gaug transform appear spont norm subgroup diffeomorph group
examin quant mot partic interact contact forc confin rectangul domain three dimend diff mass scal partic adiab sep fast slow vary perform appear berry phas magnet flux point system reduc partic aharonov bohm billiard dimend cas three dimend problem effect becom mot partic pres clos flux string billiard
pap propos algebra univers proc der fus rul baxt equ integr model wideh symmetry quant invers scat method univers baxt op certain infinit dimend repres cal oscil univers matrix wideh affin algebr first propos bazhanov lukyanov zamolodchikov quant cas also examin algebra property op
poisson algebra structures cur manifold map finit dimend rieman manifold dimend manifold investig term symplect geometry shown correspond cur manifold poisson cur algebra three gen geomet mean giv deform cur algebra geomet quant cur algebra quant cur algebra map also study
simpl manifold construct diff geometry structure investig linear connect met grav discuss comp three main approach calc result grav act funct
gaug the und quant group symmetry introduc property examin suit assum model found many simil us tim standard model spec ex four gen thu four gaug field howev class symmetry un quant symmetry theref singl coupl const rath incorp hig sect model obtain sev explicit tre level predict undeform limit weinberg angl thet boson mass fin structure const alph input also obtain predict weak coupl const mass hig break quant inv also result remain undeform gaug symmetry
effect field the typ superst the bran loc point orbifold supersymmet gaug the whos field cont conveny summ quiv diagram whos lagrang includ met coupl orbifold modul particul twist sect modul coupl fayet iliopoulo term gaug the the describ bran resolv spac spac vacu modul spac smoo met yang mil instanton whos met explicit comput instanton construct exact reproduc result kronheim nakajim
pres equ extend thom form respect vect field act framework matha quil form assocy topolog quant field the correspond twist supersymmet the cent charg analys detail diff cas topolog sigm model abel monopol four manifold
discuss transit spac heterot compact map dual transit spac typ compact calab manifold observ perturb symmetry rest wel perturb process chang numb tens multiplet least many cas simpl describ term reflex polyhedr calab manifold result suggest many perhap four dimend heterot vacu correspond typ vacu
soliton affin tod field the rel spin gen ruijsena schneider rel calogero mos model provid sought extend correspond sin gordon soliton ruijsena schneider model
complet fock spac repres cov diff calcul quant spac construct consist criter ensu algebra structure map canon ferm boson consequ algebr stat quant analys discuss conceiv stat transmut boson ferm introduc
pap direct continu begun study integr structures conform field the show construct op lambd highest weight virasoro mod commut diff valu paramet lambd op appear analog matrix baxt baxn particul satisfy fam baxt equ also show nat assum analys property op lambd funct lambd baxt rel allow der nonlinear integr equ destr veg dveg eigenvalu op equ obtain asymptot expand op larg lambd remark unlik expand op asymptot sery lambd contain dual nonloc integr mot along loc on also discuss intrigu rel vacu eigenvalu op stat transport property bound sin gordon model bas propos numb exact result finit volt charg transport point contact quant hal system
funct rel commut quant transf mat quant integr model ident class hirot bilinear diff equ equ equ complet discret class tod lat op bound standard object quant integr model ident el class nonlinear integr diff equ particul ellipt solv hirot equ giv complet eigenvalu quant transf mat eigenvalu baxt op solv auxy linear problem class hirot equ ellipt solv relev beth ansatz study nest beth ansatz equ typ model appear discret tim equ mot zero class funct bak akhiez funct determin repres gen solv bilinear discret hirot equ determin formul eigenvalu quant transf mat obtain
quant deform dimend lat gaug the defin apply reshetikhin turaev funct heegaard diagram assocy giv cel complex root un cas construct carry modul hopf algebr topolog weak coupl limit gaug the partit funct giv fold inv coincid simpl cas turaev viro discuss bound manifold wel link manifold dimend reduc obtain deform gaug the rieman surfac find connect algebra alekseev gross schomerus approach
gen funct giv numb irreduc fold eul sum poss altern sign expon sum form remark simpl obi funct equ siz search spac fold eul sum level reduc rat linear combin irreduc bas term lfloor rfloor choos analys method us tony hearn reduc achiev reduc converg doubl eul sum num method us david bailey mppslq achiev converg fold sum combin method yield bas remain search spac find confirm expect bas dirk kreim connect knot the quant field the occur perturb quant electrodynam irreduc eul sum demonst suggest transcend occ four loop contribut electron magnet mom irreduc eul sum found occ explicit analys result counterterm loop yield transcend knot numb cross
prevy pap propos construct quant group symmetry gen grav took chir vertex op the quant group cov on est ear work bas ide cov field spin repres view gen braid field exact requir transform construct conv describ grav term bloch wav coulomb vertex op thereby est first tim quant group symmetry property otim symmetry novel typ emerg cart gen eigenvalu spec cho matrix el verm mod casimir eigenvalu eq spec virasoro weight vertex op consid produc defin match condit dict hilbert spac structure op produc hid symmetry possess novel hopf lik structure compat condit root un giv right trunt linear connect prevy discuss disentangl
deform diff calcul develop bas assocy star produc dimend hamilton vect field model algebr pseudo diff op us the integr system thu obtain geomet describ op dual the also poss bas deform diff form calcul apply numb multidimend integr system hierarchy thu obtain geomet describ system limit deform disappear correspond tak dispersionless limit hierarchy
hid nonabel symmetry nonlinear interact rady mat clar term nonabel pot vary construct effect field the self induc transp phenomenon lossless coh puls prop assocy hermit symmet spac vary property self induc transp soliton numb effect pot energy gaug symmetry discret symmetry mod puls are conserv charg address elab nondeg level cas us charg conserv typ analys puls stabl giv agr ear num result
new introduc gen poisson structures bas suit skew symmet contrav tens ev ord discuss term schouten nijenhu bracket assocy jacob id express condit tens cohomolog cont giv particul determin linear gen poisson structures construct dual spac simpl algebra
find nonperturb rel langl rangl langl rangl prepot vev langl rangl supersymmet yang mil the gaug group nonlinear diff equ includ wit dijkgraaf verlind verlind equ obtain ind the ess topolog field the seen energy limit topolog string the furtherm construct relev modul inv quant der canon rel period investig structure bet funct giv explicit form modul coordin discuss uniform problem quant modul spac method propos gen supersymmet yang mil the high rank gaug group
known work feigin frenkel wakimoto typ gen fre field real cur algebr wideh assocy parabol subalgebr algebr standard cas cart borel subalgebr let obtain explicit formul wakimoto real gen cas us hamilton reduc wznw model first der poisson bracket real valu cur term symplect boson belong cur belong quant formul determin correct norm ord also show affin sugawar stress energy tens tak expect quadr form constitu
gen construct chain model certain algebra quant algebra symmetry pres symmet model giv ris sery integr system exampl chain model symmetry rel temperley lieb algebra structures repres discuss shown correspond symmet integr chain model exact solv stat discret tim resp continu tim markov chain whos spectr transit mat resp intens mat on correspond integr model
us twist real symmet group show bos ferm stat compat transform gen compact quant group drinfel typ
describ kind transit topolog distinct typ calab vacu point enh abel gaug symmetry togeth funda charg mat hypermultiplet connect appear mat loc geometry singul discuss rel instanton numb calab manifold tak part transit dual heterot string the tim process correspond higgs sem class gaug group equ vary gaug bundl spec cas situ reduc simpl conifold transit coulomb phas abel gaug symmetry
form the gen fock spac und diff form quant stat infinit bos einstein ferm dirac stat singl index system wel mult index system map singl index system study the bas three tier structure consist fock spac stat algebr gen form un vary form stat algebra also allow construct many form quant stat wel many algebra cre destruct op algebra infinit stat stat many avat consist algebr fract stat nul stat stat froz ord doubl infinit stat many repres orthost hubbard stat vary
shown triv cocyc simpl algebra us introduc antisymmet multibracket lead high ord algebra definit giv gen jacob id turn satisfy antisymmet tens high ord structure const charact algebr cocyc analys allow pres class high ord simpl algebra wel construct proc result synthes introduc singl complet brst op assocy simpl algebr
algebra hamilton structures multicompon dispersionless benney tod hierarchy study achiev us mod vary symmetry bas field symmetry en formula norm giv implicit term residu conserv charg flux calc explicit corol result equ benney tod hierarchy est shown quant express term gen hypergeomet funct simplest exampl involv legendr polynom result extend system der rat funct logarithm funct vary reduc also study
argu every calab manifold mir admit famy supersymmet toroid cyc moreov modul spac cyc togeth flat connect prec spac mir transform equ dual cyc geometry modul spac address gen framework sev exampl discuss
solv larg class hierarchy zero curv equ includ tod typ hierarchy investig hierarchy construct affin twist untw moody algebra common feat spec vacu solv correspond op lying abel cent term subalgebr interest cas subalgebra heisenberg typ us dress transform method solv orbit vacu solv construct uniform gen funct hierarchy defin altern vary correspond certain matrix el evalu integr highest weight repres definit funct apply level repres independ real vertex op particul import cas gen mkdv hierarchy wel abel abel affin tod the discuss detail
describ infinit dimend algebr hid symmetry self dual grav equ besid known diffeomorph typ symmetry affin extend infin algebr algebr contain hid symmetry affin extend lorentz rot ful symmetry algebr moody virasoro lik gen whos exponenty map solv field equ solv rel problem string the brief discuss
provid loc geomet describ charg mat ar typ the the compact calab manifold bas ide deform high singul low cart deform vary spac result agr expect bas string dual
complet program outlin pap auth migd exact fluctu around instanton solv random larg scal driv burg equ choos forc correl funct kapp exact quadr funct coordin diff result prob distribut satisfy diff equ propos polyakov without anoma term result show unless anoma term indee abs must com poss instanton solv fluctu
explicit exampl quas exact solv body problem lin pres rel hid algebr typ contain infinit many eigenst known body interact finit numb eigenst known typ deg calogero model
boy finley equ infty tod equ reduc self dual einstein equ dispersionlesslimit tod lat equ suggest dispers vert self dual einstein equ contain tod lat equ whos dispersionless limit famili self dual einstein equ system study pap result achiev us deform bas assocy star produc algebr sdiff sigm us study undeform dispersionless equ
conform field the describ dimend turb propos rec est class rat logarithm conform field the provid un candid solv resolv many drawback form approach minim model model autom includ magneto hydrodynam turb effect
demonst gen rel tod chain spec reduc dimend tod lat hierarchy reduc imply gaug equ discret akn hierarchy compon volterr hierarchy forc sem infinit vary describ unit matrix integr integr property hierarchy rev diff framework repres orthogon polynom system funct approach rel tod molec hierarchy also consid along forc apply biorthogon polynom system discuss
start loc quant field the unbrok compact symmetry group dimend spacetim construct disord field impl gaug transform field ord vary loc wedg reg enlarg loc algebra disord field obtain nonloc field the fixpoint algebra appropry extend act group shown satisfy haag dual every simpl sect spec dimend phenomenon viol haag dual fixpoint net thereby clar cas finit group extend the act upon complet canon quant doubl satisfy matrix commut rel wel verlind algebr furtherm method suit cont transp approach boson main techn ingredy strengthened vert split property hold reason mass the append part result extend arbit loc compact group method adapt chir the circ
bas algebra diff geometry method wid class multidimend nonlinear integr system obtain integr scheme equ propos
gen structure perturb expand vacu expect valu produc wilson loop op analys context chern simon gaug the wilson loop op wilson lin ord unravel algebra structure encod group fact perturb sery expand process fact theorem prov wilson lin wilson lin clos back wilson loop link inv finit typ defin integr express inv pres first three primit on low degr cas compon link addit explicit num result obtain compon link cross degr four
pap explicit prov integr system solv quant invers scat method describ pur algebra object univers matrix prop algebra repres nam exampl quant volterr model construct op funda matrix univers matrix quant affin wideh algebr oscil repres ex equ integr system symmetry algebr repres algebr
comultiply formul fus produc untw repres chir algebr gen includ arbit twist repres show formula defin tens produc suit property determin analog algebr arbit twist repres exampl study fus repres ramond sect superconform algebr lat cas certain subtl ar describ detail
pres spinon form wess zumino wit model cent approach massless quas partic cal spinon transform repres carry fract stat angl thet mult spinon stat group irreduc repres yang giv explicit result cont yang repres pres spinon cut charact formula produc obtain clos express charact hald shastry spin chain
extend study field theoret describ matrix vect model assocy many body problem dimend partic spin construct yang inv hamilton describ interact the collect boson cur algebr cub cur term ar coupl determin requir yang symmetry hamilton consist reduc finit dimend subspac stat en explicit comput spectr illust simplest cas
develop method comput excit stat energy integr quant field the iqft finit geometry spat coordin compact circ circumf iqft commut transf mat introduc conform field the gen conform iqft obtain perturb op study model fus rel transf mat trunt provid clos integr equ gen equ thermodynam beth ansatz excit stat explicit calc don first excit stat scal yang model
discuss conceiv difficul connect anticommut class ferm field argu spac class config model field describ infinit dimend supermanifold discuss main approach supermanifold examin reason many phys tend pref rog approach although berezin kost leit approach funda develop infinit dimend vary lat show funct class config consid prevy pap noth superfunct pres program fut mathem work apply class field model ferm field program part impl success pap
us supergeomet interpret field funct develop prevy pap show quit larg class system nonlinear field equ anticommut field infinit dimend supermanifold class solv construct system ar class field model us real quant field theoret model particul show suit condit smoo cauchy dat compact support isomorph correspond class solv model
counterterm reduc zet gen hypergeomet sery ar diagram triangl un rel furn insufficy dat irreduc doubl sum correspond tor knot found anom dimend larg limit comput analys term level correspond loop dimend field the loop dimend the high precid num result obtain loop us resum varepsilon expand comp analys result dimend result entail knot gen three dress prop mast loop point diagram high ord encount un posit hyperbol cross knot assocy irreduc tripl cross pair braid knot gen correspond pair irreduc doubl sum altern sign hyperbol posit knot gen self energy insert
high ord anisotrop superspac construct gen vect superbundl provid compat nonlinear connect distinct connect met structures
tors curv structure equ bianch id loc anisotrop superspac contain particul cas diff supersymmet extend prolong rieman finsl lagrang kaluz klein spac investig
gen approach form supergrav high ord anisotrop superspac contain particul cas diff supersymmet extend prolong rieman finsl lagrang kaluz klein spac giv analys three model loc anisotrop supergrav
show dimend superspac isomorph triv consist limit braid lin supersymmetry ident transl inv along lin supertransl gen cov der obtain limit quest left right der calcul braid lin
commut transf mat vertex model obey funct rel view typ tod field equ discret spac tim bas analys beth ansatz pres express solv term determin pfaff
famili nonlinear particul soliton equ ar zero curv condit connect noncommut diff calcul burg equ form col hopf transform attain interpret transform connect pur gaug mathem framework mod equ miur transform obtain joint simil set rath straightforward gen lead mod equ furtherm diff calcul assocy boussinesq equ der calcul
sery extend epstein typ provid exampl triv zet funct import phys apply regul part analys continu seen converg asymptot sery singul structure complet determin term wodzick residu gen form prov yield residu pol zet funct rightmost pol obtain dixmy trac calc ear apply pow funct analys method phys
gen rel reduc dimend possess larg group symmetry exchang class solv assocy algebr known contain affin moody algebr half real wit algebr pap exhibit ful symmetry sem direct produc wit algebr furtherm exhibit correspond hid gaug symmetry show the understood term infinit dimend pot spac involv degr freedom dilaton wel mat gravit dilaton sect linear system extend prevy known pair form twist self dual constraint analog self dual constraint ar extend supergrav high spacetim dimend result furn group theoret expl simult occur spect paramet const vary hold linear sigm model obtain dimend reduc model three dimend coupl pur grav cas algebr sem symmetry act shel field fix gaug preserv equ mot
not prov id equ ellipt gen partit funct supersymmet sigm model fold symmet produc manifold partit funct second quant string the spac tim gen funct ellipt gener shown almost automorph form context bran dynam result giv prec comput fre energy strings insid high dimend bran
system method construct manifest supersymmet dimend hierarchy pres clos express op term superfield eigenfunct obtain hierarchy equ eigenfunct equ shown autom inv extend supersymmetry supersymmet model ex lit found contain gaug equ form
discuss apply known larg expand strings supermembr strong coupl particul rec obtain mast field describ infty gaug the argu quant extrem black hol obey quant boltzman infinit stat support topolog argu strominger black hol obey infinit stat also spec form the strings bran the grothendieck mot partit funct express term funct mot beilinson conject valu funct interpret deal cosmolog const problem
shown fract superspac isomorph limit braid lin supersymmetry ident transl inv along lin fract transl gen assocy cov der emerg limit left right der calcul braid lin
defin algebr spac stat the extend supersymmetry show algebr perturb stat toroid compact heterot string clos rel gen moody algebr bran the comp form charg algebra typ compact requir string string dual find charg stat regard cohomolog class modul spac coh sheav equ algebr stat heterot dual pair elucid certain result conject nakajim gritsenko nikulin geomet defin algebra furtherm suggest nont gen algebra particul calab fold canon assocy algebra exchang mir symmetry
bas algebra diff geometry method wid class multidimend nonlinear system obtain integr scheme equ propos
renorm group method goldenfeld oono collab apply asymptot analys vect field method form bas the envelop don scal field form act complet discuss prevy work scal equ shown gen method apply equ bifurc lead landau stuart tim depend ginzburg landau equ confirm method act pow the reduc dynam reduc perturb method exampl ordin dif equ forc duff lotk volterr lorenz equ work method tim evolv solv lotk volterr equ explicit giv cent manifold lorenz equ construct simpl method
giv gen method construct complet linear independ casimir op algebr rank casimir op degr provid explicit calc symmet coefficy seen coefficy descib rat polinom rank polinom also multilinear cart algebr ind tak valu cruc point degr nee gen polinom fact rel observ whol symmet coefficy decompos subset correspond polinom cal subset clust introduc ind spec diff clust ind determin clust whatev num valu coefficy degr numb clust independ rank hent allow gen result valu rank spec gen framework explicit construct ord casimir op algebra study polinom spec num valu coefficy giv explicit
saml instanton exceiv group ar geomet collaps pezzo surfac explain phys bran prob typ compact dimend
study abel gen deform us produc polynom introduc construct explicit exampl cas provid tent quant mech describ nambu mech introduc not strong weak triv gen deform show zarisk produc nev triv eith sens exampl construct quant mech context strongly triv
found numb irreduc multipl zet valu mzvs weight gen prod enum posit knot cross posit knot field the assign knot numb mzvs first appear cross ident posit knot cross correspond irreduc mzvs virtu connect knot numb real feynm diagram loop
correl funct inv is model calc explicit us vertex op langu develop kyoto school
mech und diverg perturb the expos don detail study viol hypothes domin converg theorem lebesgu us famili techn quant field the theorem govern valid lack form manip don gen perturb sery funct integr form aspect perturb sery nee mod obtain converg sery pres us tool pract impl mod develop resum method analys light ment mech
study mir symmetry calab manifold within framework gauss manin system apply flat coordin gauss manin system period der diff equ mir addit ordin picard fuch equ period equ obtain class paramet model paramet model ferm typ calab manifold
construct vary supersymmet integr system junct pseudo diff superspac op mult compon hierarchy import particul cas obtain op sup system simil extend sup boussinesq hierarchy giv also pres minim supersymmet extend second flow hierarchy com poss integr
context batalin vilkovisky form observ abel the propos whos vacu expect valu rel alexand conway polynom three dimend cas analys explicit prov anoma fre moreov second ord perturb the formul second coefficy alexand conway polynom obtain account high dimend gen also giv
us rec adv understand perturb aspect typ strings show triv exact result quant field the reduc dual string the don construct loc geomet real quant field the togeth loc apply mir symmetry construct bas dual conject thu reduc triv quant field the result much bet understood dual typ strings moreov us principl construct system vacu structure arbit gaug the mat repres
dilut model solv interact round fac model three loc stat adjac condit encod dynkin diagram algebr regard solv vert is model crit temp magnet field theref expect scal limit govern zamolodchikov integr perturb conform field the indee rec thermodynam beth ansatz approach success unveil correspond structure certain assum nat beth ansatz solv ord check conject perform detail num investig solv beth ansatz equ crit crit model scal funct ground stat correct lowest spect gap obtain giv prec num result lowest mass ratio mass scal limit agr perfect mass ratio observ stat seem viol assum und thermodynam beth ansatz calc also analys crit spectr dilut model exhibit mass excit massless stat is conform field the
brief review energy effect the quant hal effect emphas langu famili field the model propos describ stabl hal platea jain sery mult compon abel the minim infin model lead prior class quant hal univers class expery already confirm bas predict common effect the expery soon detail property diff bas study partit funct show abel the rat conform field the minim infin model
rec rig prov nonperturb rel prepot und gaug group imply reflect symmetry overlin hold exact rel also imply invers uniform coordin modul spac quant vacu context quant symmetry point determin structure modul spac turn funct der first principl act coincid solv propos seiberg wit also consid relev gen
not polyakov equ phy veloc diff excit forc correl funct kapp alph analys sev solv cas consid good agr avail num result shown method develop polyakov apply turb short scal correl forc situ consid model self org crit
suppl discuss moor reshetikhin oth find semiclass nonabel vertex op chir antichir nonchir prim field the nonabel vertex op nat gen famili abel vertex op involv repres mat cur affin tim certain chir antichir zero mod reduc abel vertex op limit abel algebra us construct also discuss semiclass op produc expand braid rel rel known form semiclass affin sugawar conform block
deform calcul develop bas algebra structure involv grad bracket numb op left right shift op construct algebr whol structure rel algebr deform boson limit algebr root un also study detail mean chain rul expand left right der ident charg cov der encount ordin fract supersymmetry lead result op gen berezin integr fract superspac meas ar nat part form root un algebr found triv hopf structure extend assocy anyon lin dimend ordin fract superspac ident braid lin root un dimend ordin fract supersymmetry view inv transl along lin construct fract supersymmetry deform boson play rol exact analog ferm famili supersymmet cas
consid class abstract nonlinear evolv equ supermanifold model grad loc convex spac show un loc ex smooth abstract vert caus prop solv prior estim prev solv blow infinit dimend solv construct apply result class system nonlinear field equ anticommut field ar class field model us real quant field theoret model particul show suit condit smoo cauchy dat compact support isomorph correspond class solv model
describ property sem simpl assocy algebr lambd lambd grassman algebr gen show finit dimend algebr also commut hopf algebr hent finit dimend quant group select system explicit gen show rel quant envelop algebr paramet cub root un describ indecompos project repres wel irreduc on also com rel object the modul repres group bin tetrahedr group fin brief discuss rel lorentz group already suggest con mak com poss algebr mod standard model partic phys unit group sem simpl algebr assocy
investig feynm diagram calc term gen loop funct expl pres abs transcend counterterm reflect entangl link diagram construct explain unexpect rel
pres method study fibr calab manifold embed tor vary describ singl weight system analys spac ident among fibr weight giv ris three distinct typ fibr tot numb find method also study ellipt fibr fold surfac also calc hodg numb fold obtain three tim many prevy known
geometry spac group real orthogon group contract describ common point view cent extend casimir explicit giv approach trigonometry spac also adv
us method high perform comput found ind knotlik structures appear stabl finit energy soliton real dimend model explicit sim unknot trefoil config result suggest tor knot appear soliton observ op theoret poss scenario stringlike structures appear includ phys funda interact ear univers cosmolog nem liquid cryst superfluid knot soliton might act observ
canon quant wznw model provid complet exchang rel enlarg chir stat spac includ gauss compon monodrom mat regard dynam vary el lat ident satisfy diff exchang rel accord dimend the express term left right mov field autom respect monodrom inv continu rec analys problem gaug the method conclud phys stat dimend field act singl valu op inv permut coproduc left right satisfy addit constraint ful describ
construct modul inv modul spac quant vacu gaug group also introduc nonchir funct express term seiberg wit poinc met turn expect property next lead term wilson effect act whos modul property consid framework dimend regul
symmetry burg equ driv extern random forc found besid gen galil transform equ also inv tim reparamet shown gauss distribut pump forc inv symmetry break lead nont vacu instanton integr volum symmetry group provid describ fluctu around instanton lead exact solv quant mech problem
work propos geomet background the field interact strings spac high ord anisotrop approach process develop conceiv high ord anisotrop superspac un log mathem aspect modern kaluz klein the gen lagrang finsl geometry lead model phys process high ord fib bundl provid nonlinear distingush connect met structures view adopt gen field the incorp poss anisotrop stochast manifest class quant interact consequ correspond mod bas principl mathem method form phys the pres divid part first fiv sect cov high ord anisotrop superspac foc geometry distinct nonlinear connect vect superbundl consid diff supersymmet extend finsl lagrang spac analys structure bas geomet object superspac remain fiv sect devot the high ord anisotrop superst framework supersymmet nonlinear sigm model fins extend background prov energy dynam strings contain mot equ loc anisotrop field interact
giv explicit proc comput degr correl funct topolog sigm model model project fano hypersurfac homog polynom degr correl funct degr numb lin extend form cas calab hypersurfac explain polynom property preserv tool construct univers recurs formula express structural const quant cohomolog ring weight homog polynom funct const fano hypersurfac degr dimend propos conject ex form recurs formula structural const rat curv arbit degr recurs formula yield coefficy hypergeomet sery us mir calc assum valid conject find recurs law rat curv degr
describ class schwinger model study project mod algebr complex valu funct sphere mod class construct hermit connect valu univers diff envelop lead schwinger model sphere con lot program apply us hilbert spac complex inhomog form atiyah kaehl structure splits minim left id clifford algebr preserv dirac kaehl op delt induc repres univers diff envelop ord recov diff structure divid unw diff id obtain quoty us complex rham extery algebr sphere clifford act spin hilbert spac subsequ step con lot program allow defin mat act field act obtain us dixmy trac reduc integr curv squ
gen wakimoto real wideh assocy parabol subalgebr simpl algebr accord ear propos feigin frenkel pap propos mad explicit develop construct wakimoto real simpl unconv viewpoint explicit formul der wakimoto cur first poisson bracket level hamilton symmetry reduc wznw model quant perform norm ord class formul determin requir quant correct gen wideh mean commut affin sugawar stress energy tens ver expect quadr form constitu symplect boson belong cur belong quant requir cho spec polynom coordin cel flag manifold backslash effect cho investig detail construct quant coordin transform fin explicit form screening charg gen wakimoto real determin apply brief discuss
construct loc geomet model term the compact calab fourfold lead yang mil the reduc circ comput superpot funct radi gen euclid bran instanton superpot turn pot affin tod the limit van radi affin tod pot reduc tod pot
pres geomet approach field the high ord anisotrop interact conceiv high ord spac loc anisotrop spac brief spac spac introduc gen on vary typ high ord extend lagrang finsl geometry high dimend kaluz klein typ spac spin spac defin framework geometry clifford bundl provid compat nonlinear distinct connect met structures connect met discuss rel issu connect geomet aspect high ord anisotropc interact gravit gaug spin dirac spin proc field near autoparallel map introduc map deform connect extend class geodes conform transform propos vary solv problem definit conserv law spac gen background the field interact strings spac high ord anisotrop from condit consist prop clos strings high ord anisotrop background spac analys connect conform inv van renorm group bet funct gen sigm model field equ high ord anisotrop grav study detail
consid modul spac flat bundl twodimend tor real compact simpl group simply connect show connect compon describ topolog triv bundl isomorph symplect spac modul spac topolog triv bundl diff structure group phys apply isomorph allow trad topolog triv chang gaug group sketch
repres quant group cov commut cre annihil op act standard boson ferm fock spac brief address gen problem show answ posit least simplest cas
analys dimend cub schr od equ nonlinear construct cur dens rath us charg dens soliton solv found soliton mov direct rel high dimend chern simon the ind the quant result body quant problem agr weak coupl com semiclass quant soliton
found the correl funct exact solv model determin repres determin repres permit describ correl funct class complet integr diff equ barough mccoy pap show determin repres work fre ferm model obtain determin repres correl funct dag quant nonlinear schr od equ fre ferm point forthcom publ shal der complet integr equ asymptot quant correl funct model interact ferm
introduc high ord multibracket simpl algebra gen ordin algebra structure const giv algebr cohomolog cocyc virtu satisfy suit gen jacob id fin introduc nilpot complet brst op assocy multibracket algebra bas giv simpl algebr rank
gen poisson structures introduc us skew symmet contrav tens ev ord correspond jacob id giv van schouten nijenhu bracket exampl provid linear gen poisson structures construct dual spac simpl algebra
propos explicit express vacu expect valu expon field sin gordon model express agr sem class result sin gordon the perturb calc mass thir model express mak predict larg dist asymptot form point correl funct spin chain
simpl vert topolog abel gaug the construct reproduc continu express partit funct wilson expect valu link loop express term tors link numb respect feat mak poss introduc simpl field cochain assocy dual triang background manifold wel triang doubl field reminisc lat ferm doubl requir nat simpl analog hodg star op map cochain triang cochain dual triang simpl analog hodg rham the develop along nat simpl framework consid link numb fram loop loop repres tors el homolog manifold valu tors pair appear plac link numb certain discret valu coupl paramet the
find overal diverg renorm field the triv vert fulfil four term rel thu com clos est weight system provid first expl rec success assocy renorm the knot the
four loop first occ test four term rel der second auth cours investig wheth counterterm subdiverg fre diagram form weight system test rel counterterm four dimend field the yukaw interact rel prevy suspect us integr part reduc counterterm massless loop point integr four term rel ver zet zet zet demonst triv cancel trefoil knot thu support emerg connect knot counterterm transcend numb assign four dimend field the chord diagram restrict scal coupl renorm interact found necess ex pur four term rel strong ind rich structure giv fiv loop
multipl zet valu mzvs intens investig three arena knot the numb the quant field the unit kreim propos field the assign mzvs posit knot feynm diagram whos moment flow encod link diagram challeng problem pos nex knot numb field the enum posit knot enum irreduc mzvs rec tackl broadhurst kreim report larg scal analys num comput test consid sev conject numb irreduc mzvs weight dep gen prod prod frac frac shown consist shuffl id correspond it integr weight depth respect entail comput petashuffl level recount field theoret discovery mzvs counterterm eul sum gen feynm diagram success
simpl singl integr dilogarithm least tract contribut larg crit expon linear sigm model hent the spacetim dimend sol gen irreduc multipl zet valu epsilon expand epsilon sigm model epsilon the cas confirm result broadhurst gracey kreim rel knot counterterm compact form much simpl en develop term epsilon expand epsilon involv altern eul sum bas irreduc larg conclud massless feynm diagram spacetim dimend shar gre transcend complex mass diagram ev dimend contribut electron magnet mom electroweak paramet consequ perturb sect chern simon the discuss
prov determin formul quant rel problem enum sem meand nam topolog inequ plan config self intersect loop cross giv half lin giv numb point don explicit gram schmidt orthogon certain bas subspac temperley lieb algebr
detail consid maxim nonabel tod system bas class semisimpl group giv explicit express gen solv correspond equ obtain
poss ex smal correct term canon commut rel uncertainty rel rec found renew interest particul correct term could induc finit low bound delt delt resolv dist moment review gen framework path integr form quant field the gen geometry foc mech delt delt lead regul
study string the quant grav suggest ex finit low limit delt poss resolv dist latest scal planck leng within framework euclid path integr explicit show ultraviolet regul field the short dist structure rot transl inv preserv exampl geometry study detail
not optim perturb success apply energy eigenvalu gen tre wav funct quant system ingredy construct envelop perturb wav funct lead condit simil obtain principl minim sensit apply method quant anharmon oscil doubl wel pot show uniform valid wav funct correct asymptot behavy obtain first ord optim perturb ev strong coupl
review summ rec work rel form fact integr quant field the deform geomet dat assocy hyp ellipt curv rel bas deform rieman bilinear id particul lead not nul vect integr field the describ hierarchy
energy spectr three partic hamilton obtain replac body trigonomet pot sutherland problem three body simil form der express appropry vary correspond wav funct shown express term jack polynom exact solv problem three body interact explain hid symmetry
context dimend euclid grav defin nat gen dimend self dual yang mil equ dual condit curv form rieman manifold solv self dual equ provid manifold spin holonom equ eight dimend mast low dimend consid gaug field prop self dual manifold embed spin connect gaug connect solv dimend equ self dual yang mil field found show yang mil act manifold topolog bound bound sat prec yang mil field self dual result nat interpret supersymmet string the
sup grad schouten nijenhu bracket introduc us gen sup poisson structures found giv term certain grad skew symmet contrav tens lambd ev ord correspond sup jacob id express stat tens zero sup schouten nijenhu bracket lambd lambd particul cas provid linear gen sup poisson structures construct dual spac simpl superalgebra deg kil met superalgebr giv gen exampl
comput exact finit temp effect act dimend field the contain topolog chern simon term many feat common dimend chern simon the exact result explain origin mean puzzl temp depend coefficy found vary naiv perturb comput high dimend model
us coset construct comput root multipl level three hyperbol moody algebra includ bas hyperbol extend
study dimend topolog sigm model whos target spac hyp kahl manifold feynm diagram calc partit funct demonst finit typ inv manifold simil structure appear perturb calc chern simon partit funct sigm model suggest system weight finit typ inv manifold describ triv graph rieman curv play rol algebr structure const chern simon the bianch id play rol jacob id guar cal rel among weight argu spec cho partit funct sigm model yield casson walk inv gen also der walk surgery formul act finit dimend hilbert spac obtain quant sigm model dimend tor
yang doubl cent deform affin algebr addit loop paramet hbar commut rel boson quant hbar deform virasoro algebr correspond miur transform assocy screening op brst charg study moreov also construct boson typ typ intertwin vertex op fin show commut rel vertex op cas hbar act giv exact scat matrix restrict sin gordon model
pres mathem aspect landau ginzburg string vacu term tor geometry correspond tor div stat landau ginzburg model pres superpot typ typ landau ginzburg interpret tor div also pres us interpret propos method solv cal twist sect problem orbifold construct moreov construct show modul spac origin landau ginzburg string vacu orbifold connect consid mir landau ginzburg model obtain rel mor vect twist op orbifold consid en argu embed seiberg wit curv defin equ calab manifould typ string get compact rel top concern calab fourfold extrem transit discuss
random matrix the provid common mathem form distinct phys quest three diff area quant chao integr model interact calogero sutherland mos system quant grav review connect area also discuss method loop equ determin correl funct smooth glob eigenvalu correl matrix model gauss orthogon unit symplect ensembl
correl funct exact solv model describ diff equ barough mccoy pap show fre ferm cas diff equ replac integro diff equ der integro diff equ describ tim temp depend correl funct dag penet bos integro diff equ turn continu gen class nonlinear schr od equ
polysymplect form introduc analog symplect form dond weyl polymoment hamilton form field the correspond poisson bracket diff form construct analog poisson algebr shown gen commut high ord gerstenhab algebra defin text
dissip model vortex mot thin superconduc consid lagrang galil inv vert ginzburg landau model tim depend field kinet term linear first tim der field shown certain valu coupl const field dynam reduc first ord diff equ vortex posit vort circ around anoth const spee sep model
us wegn houghton equ within loc pot approxim study crit property vect model fix point togeth crit expon eigenop obtain larg valu includ infty polchinsk equ also tre peculi larg limit lin fix point pres study detail der equ pres togeth project zero mod
follow work jimbo miw introduc certain deg ellipt algebr wideh boson real investig rat limit limit cent extend yang doubl level giv boson real bas algebra reform smirnov form fact bootstrap approach sin gordon the inv thir model conject integr formul form fact sin gordon the der
analys sev issu concern singul vect topolog superconform algebr first investig typ singul vect ex regard rel charg brst inv property find four diff typ chir verm mod twenty nin diff typ complet verm mod study famy structure singul vect every memb famy map memb chain simpl transform involv spect flow famy singul vect chir verm mod follow un pattern four vect contain subsingul vect writ famy level ident subsingul vect famy singul vect complet verm mod follow infinit many diff pattern group rough fiv main kind pres particul interest thirty eight memb famy level wel complet singul vect level twenty eight diff typ fin analys orrzapf condit lead linear independ singul vect typ level verm mod writ four exampl pair singul vect belong thirty eight memb famy
highest weight typ repres the affin superconform algebra shown equ modulo respect spect flow
shown deform heisenberg algebr involv reflect op deform heisenberg algebr finit dimend repres equ repres paragrassman algebr spec differenty op guon lik form algebr rel gen stat found apply rev repres deform heisenberg algebr discuss context supersymmetry shown repres employ real dimend supersymmetry giv also poss construct univers spin linear diff equ describ eith fract spin field anyon ordin integ half integ spin field dimend
consid gapless model stat mech zero temp correl funct decay asymptot pow dist model temp correl decay expon us exampl solv model find formul describ long dist larg tim asymptot correl funct loc field formul describ correl temp arbit coupl const
lect not intoduc phys sev idea apply noncommut geometry necess mathem tool pres feel access phys illust apply yang mil ferm grav model not describ spect act rec introduc chamseddin con also pres introduc rec work noncommut lat lat us construct topolog nont quant mech field the model particul altern model lat gaug the list sect introduc noncommut spac algebra funct noncommut lat mod bundl spect calcul noncommut diff form connect mod field the mod grav model quant mech model noncommut lat append bas not topolog fand naimark seg construct hilbert mod strong morit equ part ord set pseudodiff op
definit giv phys mean quant transform commut config spac quant group coact discuss shown commut coordin transform quant group nat gen not tens op us group quant group coact induc semigroup transform stat system exampl commut transform correspond semigroup consid
univers deform heisenberg algebr involv reflect op rev shown addit wel known infinit dimend repres rel paraboson algebr also finit dimend repres paraferm nat demonst finit dimend repres repres deform paraferm algebr intern grad structure hand finit infinit dimend repres algebr supply irreduc repres superalgebr show norm form deform heisenberg algebr reflect structure guon algebr rel gen stat
wel known rec work bernard gaudin hald pasquy bghp us spin calogero sutherland model contain partic intern degr freedom respect yang symmetry follow suit mod approach bghp artic construct novel class spin model exhibit mult paramet deform nonstandard vary yang symmetry interest feat hamilton contain many body spin depend interact calc direct assocy rat solv yang baxt equ moreov spin depend interact oft lead anyon lik repres permut algebr combin intern spac partic also find gen form conserv quant wel pair ment class spin model describ method construct exact wav funct
nli linear integr equ equ beth ansatz equ finit siz gen excit stat stat hol complex root antiferromagnet ground stat consid sin gordon mass thir model period leng us light con approach model obtain continu limit inhomog vertex model nli us start point comput spectr excit stat analys larg perturb smal conform regim wel num
quant integr class mass perturb paraferm conform field the assocy compact group est show quant conserv dens scal dimend the integr valu continu vect coupl const gen perturb minim paraferm model first therm op class equ mot perturb the abel affin tod equ admit charg soliton solv whos sem class quant expect permit id exact matrix the
review rel twist supersymmet gaug the four dimend modul problem four dimend topolog study detail abel monopol equ point view relev exact result supersymmet gaug the comput topolog inv emphas background mat provid includ introduc donaldson the twist proc matha quil form
understand detail dual heterot string the compact import understand construct holomorph bundl ellipt calab manifold vary group pap develop techn describ bundl mak sev detail comparison heterot string the
pres gen scheme ident fibr framework tor geometry provid larg list weight calab fold find weight degr whos maxim newton polyhedr reflex weight degr giv ris weight project spac polynom defin hypersurfac triv canon class transvers comput hodg numb us batyrev formula der tor method first vaf fomula obtain count ramond ground stat model lat class check consist weight overlap fibr fold includ ellipt cas manifest lat follow simpl polyhedron correspond fib subpolyhedron correspond fold wherea determin bas linear project correspond fold
us form superconnect show ex boson act funct standard cyc noncommut geometry giv ris spect act principl einstein grav standard model yang mil hig term provid effect nonminim coupl boson sect lagrang
reflect equ scat lin mov half plan obtain correspond geomet pict rel config half plan touch bound plan dimend equ obtain addit tetrahedron equ consist condit mod zamolodchikov algebr
show baryon numb supersymmet twist ord coupl topolog field the abel monopol spin structures mot construct also consid aspect twist proc gaug glob cur four dimend particul rol play anoma
infinit dimend algebra hid symmetry self dual yang mil equ consid cur typ algebr symmetry affin extend conform symmetry introduc rec discuss us twist pict shown extend conform symmetry self dual yang mil equ simpl describ term ward twist construct
propos gen meand config selfintersect loop cross lin giv numb point us reform meand pair reduc el temperley lieb algebr rel quoty heck algebr nat gen also der explicit formula meand determin defin gram determin correspond bas heck algebr
seek clar phys aspect ruijsena schneider model import class model pres rel gen calogero mos model shal argu describ mislead bet simply view model paramet gen calogero mos model view model describ certain eigenvalu mot apprecy gen nat model
propos explicit express vacu expect valu bound field bound sin gordon model zero bulk mass express agr known exact result bound fre energy perturb calc
charact nambu poisson tens subfamy gen poisson on rec introduc extend ord cas discuss homolog cohomolog complex structures comp phys consid mad
loc gaug freedom rel quant mech der meas principl spac tim dirac equ obtain loc gaug transform act spin index wav funct loc symmetry allow un describ electrodynam gen rel class gaug the
propos virasoro algebr control quant cohomolog gen fano manifold determin partit funct gener construct virasoro op cas complex project spac show reproduc result kontsevich manin getzl gen instanton numb also construct virasoro op wid class fano vary cent charg algebr eq eul charact manifold
direct method obtain diff conserv law field the principl stat act propos method bas vary field funct smal loc transform spec kind act gen the rel consid
not semin giv intern sympos the el partic berlin buckow august mat der collab crem gerva klimcik part within gen framework poisson symmetry discuss approach problem construct mom map deform noeth charg gen quant group symmetry appear many conform field the concret consid cas op algebr describ liouvil the model built integ pow screenings coulomb pict
many vary calcul introduc us form braid covect algebra property certain deform paramet root un discuss detail rel fract supersymmetry spec cas dimend supersymmetry fract supersymmetry develop detail
start conform quant field the real lin show dual stil conform respect repres moebi group inf every conform norm conorm nam loc neuman algebr assocy interv coincid doubl rel commut insid loc neuman algebr assocy larg interv dual giv togeth ris infinit dimend symmetry group study class posit energy irreduc repres ment superselsect sect extend dual illust exampl gen process gen soliton sect describ fre the assocy lowest weight repres show viol regul obtain exampl moebi cov sect regul regul
show collect dynam curv domain wal dimend rel scal field model repres nambu goto membr dimend scal field defin worldsheet membr argu bas rec propos vert expand wid der expand sign reform pres purpos third four ord correct domain wal solv consid also der equ mot cor domain wal without dimend scal field equ would nonloc
pres geomet approach superst bas geomet the integr supermanifold approach provid effect calc mult loop superst amplitud arbit background mak poss calc amplitud phys stat defin brst cohomolog class us arbit repres sint form rely pres prim repres phys stat partic valu analys discret stat prim repres avail show discret stat provid inform symmetry background includ symmetry bos ferm stat dilaton exampl discret stat cov repres prim vertex form allow prov dilaton theorem direct calc
integr repres form fact expon field sin gordon model propos
us un system scheme fre field real irreduc repres construct us real correl funct sup conform model bas symmetry fre field repres gen calc fre field repres cur us determin stress energy tens cent charg model
us grad eigenvalu method rec comput extend itzykson zub integr complex mat comput point spect correl funct dirac op chir random matrix model determin diagon matrix ad obtain result scal mean level spac microscop scal find microscop spect correl funct form zero temp provid microscop vary resc temp depend chir condens
zero mod monodrom extend wznw model giv ris gaug the finit dimend stat spac gen op height cur algebr repres act dimend indefinit met spac quant group inv vect gen cohomolog dimend direct span phys subquoty
elab trigonomet vert intertwin vect fact op start point correspond ellipt construct belavin matrix naiv trigonomet limit singul car analys nee shown construct admit sev diff trigonomet deg produc quant op trigonomet ruijsena model intertwin dynam matrix obtain lat diff standard trigonomet matrix typ connect dynam matrix approach discuss
construct chern simon act deform gaug the simpl straightforward gen us spac tim continu ordin commut manifold gaug pot field strengths becom commut field approach explicit carry cas minim deform adv lead nat consist hamilton structure ess feat undeform cas exampl us poisson bracket constraint form clos algebr gen deform gaug transform
gen drinfeld sokolov form boson integr hierarchy superspac system lead zero curv form supersymmet integr system start equ superspac method symmet spac wel abel gaug techn obtain supersymmet integr hierarchy akn typ zero curv condit superspac grad algebra provid hermit symmet spac structure
propos model spliced vect bundl arbit numb gaug structures splice possess geometry split model employ connect form valu spac produc algebra theref interlac vary gaug structures triv man spec at giv structure geomet ghost sect sup algebr possess ghost emerg depend deform gaug sect assocy brst sup algebr real constraint follow inv curv
first hamilton structures recurs op connect evolv system hamilton structures supersymmet gnls hierarchy construct term superfield diff superfield bas loc evolv equ boson limit study detail loc nonloc boson ferm integr supersymmet gnls hierarchy boson counterpart der exampl cas algebr symmetry transform work rich supersymmetry structure uncov
pot gen cohomolog ring grassman giv term el symmet funct us war formul comput pow root algebra equ term coefficy consequ fus pot obtain pot explicit chebyshev polynom sev vary first kind also der fus pot reciproc algebra equ pot ident anoth chebyshev polynom sev vary display connect fus pot gen fibonacc luca numb
refin express faddeev popov determin der gaug the quant around reduc class solv apply result chern simon perturb the compact spacetim manifold quant around arbit flat gaug field isol gaug transform point prevy result finit form met independ perturb expand partit funct continu hold
pap begin describ funct method quant field the system interact partic partic obey exot stat gen col partic appear many problem condens mat phys magnet quant opt mot gen idea standard field the prov funct analog hor form wick theorem diff ord partic cre annihil op formula form express ferm boson on diff nat field allow der perturb sery the develop analog standard quant field the construct funct form
the superselect sect us consid stat loc excit vacu stat extend analys loc excit class infravacu stat appear model massless partic show correspond superselect structure stat superselect sect energy moment spectr respect vacu stat lat result obtain novel method express shap spectr term property loc charg transf cocyc find provid evid effect infravacu nat start point analys superselect structure the long rang forc
fre field repres form fact expon op affin tod model propos partic form fact calc explicit
shown dirac un defin dirac equ gen interact impos caus condit dirac der explicit formul dirac term pow sery boson pot construct extend system dirac sea system contain chir ferm caus condit yield restrict boson pot
study landau ginzburg model correspond calab four fold construct index typ stat correspond tor div index show wheth correspond div gen perturb superpot apply consid phas transit term orbifold construct obtain simpl method div gen superpot origin the gen superpot orbifold
study ex bran bound stat threshold typ string the numb situ reduc quest ex quadr study particul limit prop system bran involv der index theorem famy fredholm op support conject rel compact elev dimend supergrav typ string the show bound stat ex coincid zero bran result also provid support conject describ the matrix model addit provid evid bound stat three bran twic wrap calab van cyc
wel known anoma cancel algebr root first string revolv algebr cancel anoma princip fact lead ex heterot string fact noth ord cohomolog correspond algebra beyond ord calc seem requir spec car could study worthwhil light develop second string revolv shown rec artic algebra method bas calc casimir eigenvalu extend algebr pres artic gen irreduc repres algebr consid ord cohomolog emphas divers seen respect defin bas inv polinom term ord casimir eigenvalu alway express linear superposit easy investig inv polinim giv linear equ calc weight multipl result beyond ord includ comply though shar charact property ord calc
effect gaug field act gen charg ferm mad inv smal larg gaug transform temp suit regul dirac op determin pric par anoma resolv paradox perturb expand inv manifest temp depend induc chern simon term show larg unlik smal transform hent ward id perturb ord preserv result illust concret exampl field config
born infeld nonlinear electrodynam consid main at giv ex singul point stat field config born infeld consid model electron shown singul forbid within framework born infeld model propos modern act mak poss ex singul obtain main rel view singul init approxim model giv us linear electrodynam point charg partic
pap giv consecut describ funct method quant field the system interact partic partic obey exot stat appear many problem condens mat phys magnet quant opt mot gen idea standard field the der formula funct der partit funct green funct gen funct system exot partic lead correspond perturb sery diagram techn result illust consid dimend partic system comp exact express obtain ear
pres donaldson wit typ field the eight dimend manifold spin holonom prov stress tens brst exact met vary preserv holonom giv inv class vary sev dimend propos simil the calab threefolds manifold holonom respect point the ar consid supersymmet yang mil the defin manifold the inv met vary preserv holonom structure without nee twist stat high dimend analog fact donaldson wit field the hyp ahl manifold topolog without twist high dimend analog flo cohomolog brief outlin the ar nat within context string the
construct class quas exact solv many body hamilton arbit dimend whos ground stat correl choos known correl dimend rec novel correl high dimend reproduc spec cas spec interest exampl also writ model high dimend novel correl
shown problem calc form fact affin tod field the reduc nonperturb recurs calc polynom symmet sort vary determin recurs equ explicit sery charact polynom solv interplay weight spac und algebr repres symmet group
cohomolog compact kaehl resp hyperkaehl manifold admit act algebr resp pap show follow ide wit act follow supersymmetry particul symmetry certain supersymmet sigm model addit many funda id hodg lefschetz the also nat der supersymmetry
antiferromagnet heisenberg spin chain spin sect numb excit particul stat singl partic excit exploit fact giv simpl der bound matrix op antiferromagnet spin heisenberg spin chain diagon bound magnet field
dimend hopf algebr defin exact sequ quant group study finit quant group symmetry matrix algebr describ col sect alain con form standard model dual hopf algebr investig rec work robert coquereaux est us defin repres commut repres
quant group gen quant ant sit group particul study point view quant field the show suit root un ex finit dimend unit repres correspond ess class partic repres half integ spin structure energy class cas massless cas spin naiv repres unit fact subspac pur gaug class unit many partic repres defin correct class limit furtherm ident remark el cent play rol brst op cas root un spin assocy ghost intrins part indecompos repres show defin involv algebra cre anihil op root un exampl correspond id partic shown nonabel gaug field appear nat framework without defin connect fib bundl integr quant euclid spac sphere ant sit spac study wel giv conject us gen analys structure indecompos repres defin complet reduc assocy tens produc repres root un gen standard trunt tens produc wel many partic repres
propos paramet config spac dimend quant system id partic el symmet polynom boson ferm coordin shown paramet hamilton calogero sutherland model wel supersymmet gen express arbit valu coupl const quadr polynom gen borel subalgebr algebr superalgebr supersymmet cas algebra real first ord diff op fact est exact solv model accord gen definit giv auth imply calogero jack sutherland polynom wel supersymmet gen rel finit dimend irreduc repres algebr superalgebr
review construct multiparamet inhomog orthogon quant group project recal conjug lead quant poinc group study property univers envelop algebr giv matrix form quant algebr bicov diff calcul twist found
describ deform cov phas spac correspond cal kapp deform rel symmetry quant tim coordin heisenberg algebr obtain accord heisenberg doubl construct assocy mod uncertainty rel analys particul shown rel consist independ estim quant grav limit meas spac tim dist
geomet mean superst pict discuss detail shel gen pict chang op invers construct demonst gen op invers shel shel produc project op
der conject chir determin formula topolog algebr antiperiod algebr period algebr correspond incomplet verm mod built chir topolog prim chir antichir prim ramond ground stat respect method bas analys singul vect chir verm mod spect flow symmetry togeth comput expl consist check addit consequ uncov ex subsingul vect algebra giv exampl subsingul vect highest weight nul vect descend highest weight singul vect
origin regul approach construct spec typ symmetry bound valu problem nam renormgroup symmetry pres diff method calc symmetry bas modern group analys describ apply approach bound valu problem demonst help simpl mathem model
three partic hamilton obtain replac body trigonomet pot sutherland problem three body simil form shown exact solv writ appropry vary eigenfunct express term jack symmet polynom exact solv problem explain hid symmetry gen sutherland three partic problem includ three body trigonomet pot intern degr freedom consid analys term three first ord noncommut diff diff op construct combin susyqm supercharg el dihedr group three altern commut op also introduc
show dimend project chern simon gaug nonlinear schrodinger model equ abel gaug field the continu heisenberg spin chain the mat field geomet mean coordin tang plan spin phas spac gaug symmetry rel rot plan allow construct infinit hierarchy integr gaug the correspond magnet model inv gaug fix constraint abel the der correspond mov fram hierarchy obtain spect paramet interpret const valu stat gaug pot constrain cocyc condit
consid model infinit volum limit spin half quant spac high spin auxy spac us perturb the argu rel half infinit transf mat class model certain intertwin introduc nakayashik construct monodrom mat show spin auxy spac giv ris op
us geomet engin context typ strings obtain exact solv modul spac coulomb branch gaug the four dimend involv produc gaug group arbit numb funda mat chos pair wel arbit numb funda mat fact asymptot freedom restrict poss group funda mat chos accord affin dynkin diagram many result der el us self mir property find certain cas solv coulomb branch gaug the giv term three dimend complex manifold rath rieman surfac also study stringy strong coupl fix point ar compact high dimend the tensionless strings consid apply three dimend the
id satisfy dimend chir quant field study point view vertex algebra cauchy jacob id borcherd id three mut loc field prov consequ direct proof theorem loc system vertex op provid sev charact vertex algebra also discuss
consid matrix model mat show limit larg model describ knot diagram limit matrix string the also discuss spec prototyp atrix without matrix the ex void
consid dynam correl funct quant nonlinear schrodinger equ prevy pap found dynam correl funct describ vacu expect valu op valu fredholm determin pap show rieman hilbert problem assocy fredholm determin rieman hilbert problem form permit writ complet integr equ fredholm determin perform asymptot analys correl funct
princip bundl consid subspac spac connect tang bundl giv tang bundl spac connect tang gaug group act fre appropry brst op introduc quant field the includ field el wel tang vect spac curv simplest apply brst symmetry cal yang mil the describ relev gaug fix condit analys brief account topolog the also includ relev batalin vilkovisky op describ
us describ dimend einstein grav chern simon the extend paramet famy describ einstein grav don replac poinc gaug group symmetry deform poinc gaug group symmetry form recov result obtain paramet famy hamilton form grav although form term noncommut dreibein spin connect field express act field equ appropry ord id form ordin on moreov start prop defin met tens us met the built christoffel symbol spac tim curv us express term met tens repres numb artic also coupl the partic sourc find sourc carry exot angul moment fin problem rel introduc cosmolog const discuss
jacob bracket gen standard poisson bracket leibn rul replac weak condit extend bracket involv arbit ev numb funct structure includ particul cas rec introduc gen poisson structures linear cas simpl group manifold also study triv exampl diff com gen poisson structures construct found us cohomolog ring giv group
expl describ chir block term inv der verlind formul model obtain entir bas repres the affin algebra contrast ex proof verlind formul approach work univers untw affin algebra produc obtain homolog interpret verlind multipl eul charact complex built inv tens finit dimend simpl algebra result also us comput certain trac automorph spac chir block argu rig pres form pap theref submit publ
phenomenon class quant discuss rev class pseudoclass gaug system nonlinear nilpot constraint contain fre paramet vary paramet chang loc gaug discret symmetry correspond system spec discret valu giv ris maxim glob symmetry class level exact valu paramet sep quant level particul singl requir conserv discret symmetry phenomenon observ famili pseudoclass model inv mass ferm system pseudoclass model inv system topolog mass gaug field
start work hypothes phys correspond mathem describ mean discret conceiv planck scal many problem fac enterpr find discret protoform build block continu phys mathem cor conceiv not dimend follow develop not irregul structures lik larg graph network der numb property among thing show stabl wid class perturb import dimend phas transit mind furtherm system construct graph almost arbit fract dimend context dimend renorm stat mech irregul set
discuss apply method angul quant reconstruct form fact loc field mass integr model gen form illust exampl klein gordon sinh gordon bullough dod model lat model angul quant approach mak poss obtain fre field repres form fact expon op discuss intrigu rel fre field repres deform virasoro algebr deform assocy bullough dod model appear diff known deform virasoro algebr
us supergeomet interpret field funct show class class field model us real quant field theoret model infinit dimend supermanifold class solv minkowsk spac construct show smoo cauchy dat compact support isomorph correspond class solv model
consid euclid bran wrap around manifold exceiv holonom dimend sev eight result the bran dimend reduc dimend supersymmet yang mil the cohomolog field the describ topolog modul spac instanton dimend the bal cohomolog the giv act pot chern simon typ produc method construct rel cohomolog field the describ monopol modul spac manifold holonom
perturb dirac first ord extern pot calc expand around light con shown perturb consist caus contribut describ singul behavy dirac light con contain bound lin integr pot part der caus contribut smoo funct prep step construct form solv inhomog klein gordon equ term infinit sery lin integr gen method pres us explicit analys feynm diagram dirac klein gordon wav equ posit spac
form scheme pres einstein class the gen rel appear common inv sect paramet famy diff the achiev replac poinc group ordin tetrad form deform poinc group us the recov although writ term noncommut vierbein spin connect field the met sect lead ordin einstein hilbert act correspond equ mot christoffel symbol compon rieman tens ordin commut numb us form term met tens built appropry bilinear vierbein furtherm exhibit paramet famy hamilton form gen rel show canon form ashtek built valu constraint stil polynom poisson bracket skewsymmet diff
giv explicit proof within framework beth ansatz string hypothes fact multipartic scat antiferromagnet spin heisenberg spin chain cas partic
prov modul spac stat solv model spacetim sigm sigm compact rieman surfac geodes incomplet respect met induc kinet energy funct geodes approxim predict theref lump collaps form singul finit tim model
investig exact result isotrop turb three dimend press grady neglig der exact point correl funct dens three dimend show dens dens correl behav alph alph frac sqrt shown three dimend energy spectr inert rang scal expon frac sqrt simeq also discuss tim scal exact result valid strong turb pres press confirm predict us rec result num calc expery
expl rel black hol jackiw teitelboim dilaton grav soliton sin gordon field the analys expand wel known connect solv sin gordon equ const curv met particul show solv dilaton field equ giv met the also solv sin gordon equ linear correspond soliton sint dilaton gen kil vect const curv met interest analog interpret term symmetry soliton solv also show ddot cklund transform rel diff soliton solv poss construct flat connect form bas gaug the form dilaton grav
us tool diff calcul quant algebra assocy quant group find paramet famy gaug the assocy quant group although gaug field spin connect vierbein commut object depend deform paramet poss construct met the insensit deform christoffel symbol rieman tens ordin commut object hent argu torsionless einstein gen rel common inv sect paramet famy deform gaug the
examin structure spacetim symmetry toroid compact string the within framework noncommut geometry follow propos frohlich gawedzk describ noncommut string spacetim us detail algebra construct vertex op algebr show spacetim dual discret worldsheet symmetry string the consequ ex independ dirac op ar chir structure conform field the demonst dirac op also respons emerg ordin class spacetim energy limit string spacetim est rel dual chang spin structure target spac manifold study automorph group vertex op algebr show spacetim dual nat gaug symmetry form show class gen cov also becom gaug symmetry string spacetim expl larg symmetry algebr context univers gaug group string the connect symmetry group algebra structures ar mathem the vertex op algebra monst group also brief describ class topolog spacetim mod string the calc cohomolog group noncommut spacetim self contain pedagog introduc techn noncommmut geometry also includ
canon quant the fre field us arbit foly flat dimend spacetim investig shown dynam evolv along arbit spacelik foly unit impl fock spac assocy inert foly follow schrodinger pict ex arbit foly unit im heisenberg pict the explicit construct schrodinger pict im heisenberg fock spac stat provid result pres interpret term dirac constraint quant paramet field the particul shown schrodinger pict phys stat satisfy funct schrodinger equ includ slic depend numb quant correct accord propos kuch spat diffeomorph inv schrodinger pict phys stat est funda difficul ar try gen result high dimend spacetim
continu prevy work develop certain piec funct analys gen graph cre con cal spect tripl hilbert spac structure repres certain funct algebr socal dirac op encod part geomet algebra property graph der particul explicit express con dist funct show gen bound ordin dist graph typ strictly smal lat exhibit among thing und reason phenomenon
study detail sem infinit brst cohomolog gen affin algebra cohomolog relev brst approach gaug wznw model prov ex infinit sequ el cohomolog zero ghost numb imply brst approach topolog wznw model admit many stat conv coset construct conclud also apply topolog model work also contain result structure verm mod affin algebra particul gen result verm bernstein fand fand finit dimend algebra structure multipl verm mod
overview quant integr system pres bas conceiv the highlight stressing un algebra property help gen system repres op diff model also solv rel eigenvalu problem almost model independ diff approach integr ultraloc nonultraloc quant model explain interrel subject focuss
comput mean beth ansatz bound matrix op anisotrop spin chain diagon bound magnet field noncrit regim delt result form term gamm funct agr vertex op result jimbo
ferm boson ghost system defin term singl vertex algebr admit paramet famy conform structures observ structures rel provid simpl obtain charact formula gen twist mod ghost system symmetry subgroup und twist mod also defin infinit inv vertex subalgebra structure study detail algebr point view particul emphas inv subalgebra ferm ghost system
exact expect valu field bullough dod model der adopt reflect rel involv reflect matrix liouvil the wel spec analys assum us result propos explicit express expect valu prim op minim perturb op result concern perturb minim model also pres
method der solv nahm equ bas root structure simpl algebra giv illust method rec found solv nahm equ tetrahedr octahedr symmetry shown correspond root system
artic interact topolog phys appear book cal hist topolog edit io jam publ elsevy nor holland
semiclass approxim partit funct chern simon gaug the der us inv integr method volum scal fact undetermin fix hand prevy der autom tak account framework agr wit exact express partit funct weak coupl larg limit ver gaug group spacetim
prevy artic propos virasoro algebr control quant cohomolog fano vary gener pap construct fre field describ virasoro op quant cohomolog shal show ev homolog class hler manifold fre boson ferm field virasoro op giv simpl bilinear form field shal show virasoro condit correct reproduc gromov wit inv also cas manifold van analys class suggest virasoro condit hold univers compact smoo hler manifold
talk rel black hol jackiw teitelboim dilaton grav soliton sin gordon field the describ wel known connect solv sin gordon equ const curv met review expand particul shown solv dilaton field equ giv met the also solv sin gordon equ linear correspond soliton next shown ddot cklund transform rel diff soliton solv poss construct flat connect form bas gaug the form dilaton grav fin supersymmet gen sin gordon the review lead approach problem origin dimend black hol entrop
lect pres gen introduc topolog quant field the the discuss framework matha quil form context twist supersymmet the discuss detail rec develop donaldson wit the obtain apply result bas dual supersymmet yang mil the involv describ comput donaldson inv term seiberg wit inv gen donaldson wit the review structure vacu expect valu observ analys context dual simplest cas
analys plan contribut donaldson inv four manifold contribut van donaldson inv must writ plan integr contribut plan integr quit int analys gre detail ev calc analys plan integr rel donaldson the supersymmet yang mil the describ much ful rel donaldson inv the gen four manifold simpl typ interest formula obtain class numb imagin quadr field also show result gen extend donaldson the obtain includ hypermultiplet mat field
rol reduc connect abel seiberg wit inv analys massless topolog monopol extend abel group vert giv smal extern field found vacu expect valu sep part donaldson the part abel monopol the part abel monopol the produc find id topolog inv proof dual rel hig mech necess
study model quant stat mech solv algebra beth ansatz gen method calc correl funct bas method determin repres auxy fock spac auxy bos field introduc ord remov body scat repres correl funct mean valu determin fredholm integr op repres simpl form larg spac tim sep pap explain calc mean valu auxy fock spac asymptot express fredholm determin necess evalu asymptot form phys correl funct
investig class limit knizhnik zamolodchikov bernard equ consid system station schr od equ singul curv tim modul curv form reduc autonom hamilton system includ particul exampl schlesinger equ painlev equ gen gen cas defin hierarchy isomonodrom deform respect chang modul underl curv accompany whitham hierarchy phas spac spac flat connect bundl addit dat mark point der fre field the hamilton reduc act gaug symmetry subsequ fact respect diffeomorph curv approach allow defin equ assocy linear system whos isomonodrom deform provid addit lead describ solv project method spec limit convert hitchin system particul bundl ellipt curv mark point obtain limit ellipt calogero body system
canon structure spac tim symmet analog hamilton form field the bas dond weyl the study spac tim dimend polymoment assocy spac tim der field vary polysymplect form gen simplect form giv ris horizont form play rol dynam vary vert multivect gen hamilton vect field grad poisson bracket defin form lead structure grad algebr subspac cal hamilton form ex gen poisson structure ar form cal high ord right gerstenhab algebr field eu equ mot form form term grad poisson bracket hamilton form spac tim volum form hamilton funct apply scal field electrodynam nambu goto string rel standard hamilton form field the brief discuss detail improv account ear cont commun
gen poisson algebr field the canon form term polymoment vary discuss grad poisson bracket diff form bracket funct consid poisson bracket diff form giv ris vary gen gerstenhab algebr noncommut sens loday high ord sens high ord grad leibn rul bracket fulfil property nambu bracket includ funda id thu lead nambu poisson algebr point field the context nambu bracket prop defin cov analog hamilton funct determin joint evolv sev dynam vary
gen famili eul equ three dimend propos admit sufficy larg numb conserv law permit integr quadr us class analog nahm equ exampl discuss sev dimend eul ar class counterpart eight dimend self dual equ cur believ play rol develop string the
four dimend gaug the ex wel known correspond instanton holomorph curv simil correspond ex certain octon instanton triholomorph curv prov lat correspond stem dynam vary dimend reduc dimend supersymmet yang mil the prec show dimend reduc dimend supersymmet sigm model hyperkaehl otherw arbit target four dimend hyperkaehl manifold topolog sigm model loc spac triholomorph map hyperinstanton modul spac instanton four dimend hyperkaehl manifold the interpret term supersymmet gaug the cas topolog sigm model understood adiab limit dimend reduc dimend supersymmet yang mil eight dimend manifold holonom spin cohomolog the loc modul spac octon instanton
fre field repres typ vertex op corn transf mat eight vertex model propos construct us vertex fac correspond mak poss express correl funct eight vertex model term correl funct model nonloc insert nonloc insert admit fre field repres term lukyanov screening op spectr corn transf matrix baxt kelland formul av stag pol reproduc
discuss random matrix model term el op blu funct funct invers green funct show op embody ess numb phys phenomen wheth away crit point illust assert borrow numb rec result effect vacu mat provid simpl phys argu fav univers continu spect oscil wheth zero virt bulk spectr chir crit point also discuss effect quant system disord strong weak dissip hatano nelson loc
propos integr supersymmet tod lat hierarchy relev construct supersymmet matrix model defin first hamilton structures recurs op pair repres provid part evid ex infinit dimend superalgebr flow study boson limit introduc pair repres boson tod lat hierarchy fin discuss relev approach construct supersymmet gen tod lat hierarchy
jackiw teitelboim gaug form dimend grav allow rel diff gaug fix condit integr hierarchy evolv equ show equ zweibein field writ pair tim revers evolv equ react diffus typ admit dissip solv spect paramet rel pair appear const valu spin connect assocy gaug symmetry spont break compact symmetry irrevers evolv discuss
zero curv repres dimend integr model gen spacetim dimend introduc form connect gen zero curv condit us repres equ mot rel inv field the phys interest dimend the chern simon grav model dimend self dual yang mil the bogomolny equ approach lead method construct conserv cur solv submodel dimend model explicit construct infinit numb prevy unknown nont conserv cur posit integ spin repres construct conserv cur lead lorentz scal charg
find gen hyp ellipt solv compon reduc nahm equ propos hitchin ellipt solv spec cas appear spec valu monopol charg
pres novel form instanton equ dimend yang mil the form rev equ last memb sery gaug theoret equ assocy real divid algebra includ flat dimend ant self dual us form prov flat spac equ understood term mom map spac connect modul spac solv obtain gen symplect quoty kaehl quoty dimend hyperkaehl quoty dimend octon kaehl quoty dimend extend equ curv spac wherea dimend equ mak sens surfac dimend equ mak sens arbit ory manifold dimend equ mak sens manifold whos holonom contain spin interpret equ term mom map constraint manifold surfac must ory manifold must hyperkaehl manifold must flat
analys structure perturb sery expand chern simon gaug the light con gaug introduc regul prescrib entail consid fram knot pres gen form vacu expect valu wilson loop result express turn giv fram depend obtain us perturb method perturb method cov gaug also contain kontsevich integr vassiliev inv fram knot
giv extend casimir casimir algebra includ vertex op depend simpl root
rel entrop dimend field the study apply irrevers quant renorm group rely gen monoton theorem quant prevy est cylind geometry interpret finit temp field the defin rel entrop monoton quant simil zamolodchikov funct hand dimend quant thermodynam entrop also lead monoton quant diff property rel thermodynam quant complex compon stress tens also est hent entrop theorem propos analog zamolodchikov theorem cylind geometry
pres hermit matrix chain repres gen solv hirot bilinear diff equ three vary larg limit matrix model provid explicit particul solv continu diff hirot equ three vary rel repres eigenvalu transf mat quant integr model discuss
rec show post diffeomorph equ stat imply quant mech approach tak canon vary depend rel part exploit bas symmetry und canon form particul look spec transform lead fre system van energy furtherm hand equ principl consist impl class mech nat quant analog hamilton jacob equ thu imply schroedinger equ let show act principl un lead solv furtherm find reduc system fre van energy der transform leav wav funct inv also express canon schroedinger equ mean bracket rec introduc framework bracket analog poisson bracket canon vary tak depend
pap giv inh tor describ spec class sheav known equ sheav tor vary part klyachko apply technolog heterot compact particul model distl kachru also discuss knowledg equ sheav us reconstruct inform entir modul spac sheav many result relev heterot compact prevy known mathem collect exampl result concern wheth restrict stabl sheaf calab hypersurfac remain stabl stat also describ substruct kahl con modul spac sheav independ kahl class within subcon study the compact light fact also discuss seen context equ sheav tor vary fin brief spec apply result mir symmetry
obtain symmetry algebr mult matrix model plan larg limit algebr assocy matrix model quant spin chain particul certain mult matrix model exact solv us known result solv spin chain system
pap pres construct isospect problem tor construct start vert gaudin model rec study kurok takeb context twist model class limit quant hamilton gen gaudin model turn class hamilton nat matrix structure hamilton us build autonom mult tim hamilton system ev shown isomonodrom problem tor isomonodrom problem also reproduc ellipt analog equ twist model fin geomet interpret isomonodrom problem discuss langu modul spac meromorph connect
explicit express effect act simpl group brok subgroup est simpl direct form simil act chern simon form explain apply also discuss
continu work explicit formul point funct dimend inv is model found obtain diff result high correl funct sev consist check don
form fact sequ integr defin axiom solv system recurs funct equ known form fact equ show solv replac study repres the novel algebr assocy giv partic matrix follow feat contain doubl algebr subalgebr form fact ar therm vect stat temp therm ground stat correspond loc op und finit temp structure indirect rel unruh effect rindl spacetim manifest modul structures delt sens algebra impl explicit term gen
der exact equ mot rel vortex three dimend model complex field veloc giv term grady complex field vortex posit discuss problem reduc field dynam clos dynam system loc interact strings funda degr freedom
find microscop spect dens spect correl assocy multicrit behavy hermit complex matrix ensembl show univers conject microscop spect dens dirac op certain the without spont chir symmetry break belong univers class
giv intrins definit spec geometry ar glob supersymmetry four dimend bas algebra integr system exhibit geometry integr hypothes spec kahl manifold rel integr system cotang bundl spec kahl manifold carry hyperkahl met also defin spec geometry supergrav term spec geometry glob supersymmetry
shal pres proof univers microscop spect correl verbaarschot random matrix model corrob beauty agr predict gauss model num dat rath follow clos origin proof shal disgu conv approach op method employ altern proof kanziep freilikh hop borrow not quant mech proof pedagog flav shal also discuss problem assocy mult crit
analys giv explicit repres effect abel vect gaug field act gen charg ferm particul at therm regim dimend spect asymmetry pres show zet funct regul smal larg gaug inv preserv temp numb ferm us pric anoma hel par inv lost ev dimend lat ev zero mass gaug inv dict gen foury repres act term holonom carry novel larg gaug inv inform show larg unlik smal transform hent ward id perturb ord preserv clar rol prop redefin chern simon term context pow repres act term massless heat kernel abl obtain rig gaug inv expand smal larg ferm mass sep par ev part arbit dimend repres also display nonperturb origin finit renorm ambigu phys resolv requir decoupl infinit mass fin illust gen result explicit comput effect act phys exampl field config three dimend cas conclud finit temp effect phys relev nonabel result pres sep
study born infeld equ lagrang point view emphas dual symmetry pres system obtain hamilton form direct lagrang also show system admit dispersionless nonstandard repres direct lead conserv charg includ on prevy obtain also pres gen funct charg point vary property system
pres subsingul vect superconform algebra on becom singul chir verm mod report rec gato river rosado show larg class singul vect topolog algebr becom subsingul vect antiperiod algebr topolog untw class consist brst inv singul vect rel charg zero conform weight label singul vect turn result subsingul vect transform spect flow subsingul singul vect period algebr writ singul subsingul vect start topolog singul vect level
discov gaug inv observ matrix model inv form algebr plan larg limit model includ quant chromodynamics atrix the strings study gaug inv stat correspond op strings meson find algebr extend remark algebr lambd produc wel known algebra infty cuntz algebr lambd appear gen algebr vect field circ commut geometry also repres algebr est isomorph certain matrix model preserv gluon numb op quant spin chain us known result quant spin chain abl ident exact solv matrix model fin hamilton dimend reduc model express explicit el algebr
class explicit construct embed diagram verm mod supersymmet extend virasoro algebr ess ingredy solv consist draw distinct diff typ submod appear verm mod problem simpl assocy every verm mod relax verm mod affin algebr isomorph embed diagram mak mech accord structure relax embed diagram found know standard embed diagram verm mod result class relax embed diagram follow pattern extend addit ind numb twist standard embed diagram contain giv relax embed diagram
construct neveu schwarz superconform field the giv superfield form also review result definit superconform manifold gen conto integr tayl expand superconform spac arbit defin uncharg prim field giv infinitesim chang superconform transform lead op produc expand stress energy tens prim field der wel known commut rel neveu schwarz superconform algebra context observ cent extend term disappear neveu schwarz the fin giv glob transform rul prim field act algebr gen
exact perturb approach us der el correl length rel mass gap dimend dilut lat model regim beth ansatz solv regim model lat real dimend is model magnet field calc model regim start eight thermodynam sign string typ found prevy num study string typ seen consist ord high field limit eight mass obtain reduc approach crit mass predict zamolodchikov thu provid direct lat determin mass spectr
construct gaug inv op singul knot context chern simon gaug the op provid polynom inv vassiliev inv singul knot apply pres form kontsevich integr cas singul knot
survey cohomolog phys phrase first appear context anoma gaug the diff form implicit phys least back gauss electro magnet definit link numb vis dirac magnet monopol magnet charg giv first chern numb thu charact class imply cohomolog algebra group introduc phys ghost introduc fad popov incorp cam known brst cohomolog lat ghost reinterpret gen chevalley eilenberg complex algebr cohomolog cohomolog phys also mak group theoret cohomolog algebra deform the espec novel extend homolog algebr combin algebr cohomolog kosz tat resolv maj emphas talk synerg combin kind cohomolog appear batalin fradkin vilkovisky approach cohomolog reduc constrain poisson algebra analog vert develop batalin vilkovisky approach quant partic lagrang lagrang string field the revid view batalin vilkovisky machinery recogn part reconstruct homolog algebr pow idea undreamt disciplin
study spectr scal yang model finit interv point view gen trunt conform spac approach system bound bound thermodynam beth ansatz allow reflect fact match spec bound condit lead propos minim famy reflect fact describ relev bound perturb model equ propos prevy ground stat interv must rev certain regim find necess mod analys continu also propos equ describ excit stat check equ bound trunt conform spac dat access finit siz spectr en observ bound flow bulk remain massless form bound bound stat bulk mass
illust vary way algebra framework noncommut geometry nat capt short dist spacetim property string the describ noncommut spacetim construct vertex op algebr show algebra property bear striking resembl structures appear the noncommut tor class in automorph spac show nat imply conv dual symmetry quant geometry spacetim examin problem construct univers gaug group ov dynam symmetry string spacetim also describ aspect toroid compact light lik coordin show certain gen moody symmetry monst sporad group ar gaug symmetry result spacetim superst the
coupl linear integr equ der class model connect quant group simply lac algebr solv us beth ansatz equ describ arbit excit stat system finit spat leng gen destr veg equ sin gordon mass thir model affin tod field the imagin coupl const apply cent charg conform weight conform field the extract straightforward man quant group trunt root un discuss detail limit recov proc rcfts extend conform symmetry
comput gen correct integr hierarchy describ coupl grav topolog field the bihamilton structure hierarchy giv class algebr comput cent charg algebr also express gen funct ellipt gromov wit inv funct isomonodrom deform problem ar the wdvv equ assocy
study rel group algebra approach dress symmetry soliton solv tod field the dimend origin soliton affin tod model found ol turok underwood singl soliton cre expon el diagon princip heisenberg subalgebr altern babelon bernard exploit dress symmetry reproduc known express funda funct sin gordon model pap show equ method construct soliton tod model
geroch group isomorph affin moody group infinit dimend solv gen group einstein equ surfac orthogon commut kil vect introduc anoth solv gen group equ dress group discuss connect geroch group show act transit dens subset modul spac pair express twist self dual system study dress problem assocy also describ vertex op solv reduc einstein equ particul allow find solv pur algebra comput
construct topolog recurs rel high gener gen dimend topolog field the coupl grav combin virasoro condit en determin numb high gen holomorph curv fano vary cas reproduc known result gen
consid quant sinh gordon model pap us known formula form fact contribut obtain clos express correl funct express determin integr op simil determin repres prov us the correl funct also matrix model
modul heterot compact ellipt wel dual theoret compact break bas paramet ev nat involv ellipt curv fib twist paramet lat includ continu part wel discret part interpret heterot modul term cohomolog group spect cov ident correspond theoret modul certain stabl deg argu bas comparison three geomet object spect cam cov pezzo fibr continu part twist modul amount isomorph certain abel vary connect compon heterot prym vary mod jacob theoret intermedy jacob comparison discret part gen match heterot bran theoret bran imp
dual consid perturb the der giv sery solv smal paramet dual sery develop paramet invers dual symmetry perturb the ident shown dual dyson sery quant mech giv rec dev sery adiab approxim lead ord simpl apply result giv red theorem strongly perturb quant system
gen solv analog eul equ shown giv integr rieman surfac gen model der spin inv self dual yang mil equ depend upon vary regard model describ self dual membr instanton sev integr reduc low target spac dimend discuss giv descend eul assocy rieman surfac gen respect
est second quant solv nonlinear schrodinger equ half lin mix bound condit solv bas algebra structure cal bound exchang algebr substitut pres bound famili zamolodchikov faddeev algebr
project wilson polchinsk renorm group equ onto uniform extern field depend effect fre energy connect green funct result hierarchy equ admit cho nat trunt clos schemes nonperturb approxim solv approxim schemes gen avoid pow sery expand eith field moment follow clos scheme lowest ord equ mean field approxim anoth clos scheme giv loc pot approxim extend clos schemes high ord lead interest quest regard trunt schemes converg nonperturb approxim scheme bas novel moment clust decomposit connect green funct seem off poss acc nonperturb success approxim
the compact lead consid manifold obtain rol calab threefolds calab modul spac result spac gen singul govern extrem transit undergon provid background simplest cas conifold transit study employ topolog method smal class theorem smoo simply connect spin clos manifold comput cod appendix understand manifold appear link singul sing calab threefolds typ primit contract smoo obtain many loc admiss extrem transit pair calab threefolds list glob real requir study mathem byproduc pursuit subject obtain formul comput topolog bound tubul neighb gorenstein rat singul pezzo surfac embed smoo calab threefold div
consid scal yang model correspond un perturb minim model unit model us known express form fact ord obtain clos express correl funct trac energy moment tens express determin integr op simil determin repres prov us quant correl funct also matrix model
discuss pap behavio minim model conform the perturb op bound us rso restrict sin gordon model adapt bound problem sery bound flow diff conform inv bound condit describ gen staircas phenomenon discov zamolodchikov find analys continu bound sinh gordon model provid flow interpol minim model bulk also poss conform bound condit particul cas bulk sinh gordon coupl turn zero obtain bound roam traject the interpol poss spin kondo model
heterot string compact dimend ellipt fib calab conject dual the compact dimend calab fib bas ellipt fib particul modul the isomorph cas relev phys compact dimend respect mathem richest pict seem emerg modul spac involv analys integr system whos fib admit rath diff describ the purpos talk review known known conject isomorph und mathem princip bundl ellipt fibr review accompany taniguch talk
talk discuss describ modul spac princip bundl ellipt fibr term cam cov distinct prym vary emphas clos rel problem integr hitchin system gen discuss rough parallel addit exampl includ import step argu illust apply heterot the dual describ accompany icmp talk
study dynam symmetry break three dimend chern simon term consid screening effect flav ferm find phas vacu ferm mass magnet field dynam gen coefficy term kapp eq result vacu becom finit dens stat half fil ferm kapp find ferm remain massless magnet field induc kapp spont magnet occ regard extern field
investig self dual yang mil gaug config tim gaug symmetry brok wilson loop construct explicit field config singl instanton nahm method show instanton compos self dual monopol opposit magnet charg norm modul spac met instanton study vary limit field config modul spac met
sint subject noncommut geometry ent mat felt nee pres mat undergradu cours level review zero ord approxim project thu pres pap attempt off mot mathem prerequisit deep study least serv support glant rec result theoret phys
problem gaug clos form consid target manifold simpl group seen obstruct gaug subgroup subset construct form cocyc rel algebr cohomolog equ cohomolog explicit gen express cocyc giv common geomet structure gaug clos form hok weinberg effect act typ wel obstruct ex also exhibit explain
typ string the compact calab threefold believ dual heterot string surfac tim tor surfac consid result modul spac hypermultiplet ident pict cas heterot string exampl discuss bundl bundl surfac cas point lik instanton lead rath beauty id integr cohomolog calab threefold integr structures heterot sid somewh reminisc mir symmetry discuss consequ prob nonperturb effect typ string heterot string
topolog superconform algebr diff typ singul vect complet verm mod distinct rel charg brst inv property vect prim built wherea typ ex level zero remain typ ex gen level construct already level pap writ map typ topolog singul vect singul vect antiperiod algebr result obtain construct formula typ topolog singul vect us construct formula singul vect doerrzapf
cas prop unit zero mass finit part scal loop tetrahedr vacu diagram reduc let word let alphabet form omeg omeg lambd lambd six root un three diagram yield zet omeg omeg cas combin eul zagy zet omeg omeg omeg three cas combin squ claus zet omeg omeg cas mass involv const mass delign eul zagy appear zet omeg omeg omeg prevy unident term loop paramet standard model mer zet remark simpl result stem shuffl algebra nest sum it integr diagram evalu digit second primit word transform expon converg singl sum rec shown zet zet famili const whos bas sup fast comput mass involv novel bas diagram reduc const produc mass cas entail bas
pap giv quick account simplest cas hitchin integr system knizhnik zamolodchikov bernard connect gen particul construct act angl vary gen hitchin system group exploit rel class neuman integr system
construct asymptot loc flat gravit instanton modul spac solv nahm equ allow find twist spac kahl pot
consid clos op integr antiferromagnet chain construct inv matrix clos chain extend analys sutherland kul reshetikhin consid also complex string solv beth ansatz equ solv ess describ gen multipartic excit stat also explicit determin quant numb stat particul model partic lik excit funda repres direct comput complet partic mat cas op chain diagon bound field show transf matrix symmetry wel dual symmetry interchang help symmetry comput mean beth ansatz partic typ correspond bound mat
model asymptot loc flat gravit instanton modul spac solv bogomolny equ prescrib singul study modul spac us ward correspond find twist describ en writ ahl pot gravit instanton rel explicit form
show ory surfac poss bound gen kram wanny dual world quant group gen mot quant poisson dual string the cohomolog quant coefficy defin surfac mean rev funct respect glu op connect inv fold
plan integr topolog twist supersymmet gaug the gen contain contact term nonloc topolog observ pap propos interpret contact term point view integr hierarchy whitham deform inspir mar moor remark blowup formul plan integr contain piec interpret singl tim funct integr hierarchy singl tim funct extend mult tim vert without spoil modul inv blowup formul mult tim funct compr gauss fact thet funct tim vary play rol phys coupl const observ carry exceiv div coefficy gauss part ident contact term observ id examin langu whitham equ relev quant writ form der prepot
larg class finit algebra defin rel yang prov satisfy theref finit algebra appear real yang result us determin property algebr repres
construct algebr homomorph yang finit algebra show result apply determin property finit dimend repres algebra
rec pap oht townsend study condit must satisfy config intersect bran angl supersymmet pap extend result numb bran numb bran accompl interpret result term calibr geometry independ interest
exploit rel fredholm mod seg shal stinespr vert canon quant tak start point first quant field describ con axiom noncommut spin geometry hamilton framework ferm quant field noncommut manifold introduc analys ultraviolet behavio second quant field noncommut tor discuss behavio expect noncommut spin manifold
consid supersymmet gaug the imp vary dimend system ar study intersect bran unlik conv gaug the hig branch imp the compact direct model eight supercharg hig branch hyperkahl manifold giv modul spac solv certain diff equ equ dimend reduc self dual equ bound condit determin imp also interpret nahm transform self dual equ toroid compact spac discuss apply result light con form yang mil the solv certain gaug the
consid analog vertex model construct altern spin spin lin ident transf matrix spac act term repres the diagon transf matrix comput matrix giv trac formul loc correl funct point funct spin loc vary altern lat particul ground stat giv linear combin point funct pur spin model diff ground stat mix ratio calc exact express term irreduc charact deform virasoro algebr
study coulomb branch plan integr supersymmet yang mil the manifold us wal cross argu der express donaldson inv manifold explicit express tim rieman surfac gen obtain us kroneck doubl sery id result might us fut study quant cohomolog
show phys quant sev dimend condens mat model rel seiberg wit calc exact quant supersymmet gaug the particul magnet kondo problem cur bound sin gordon model express form exampl lat rel bound mass scal analog lambd proport radi boson squ thu ir curv infinit gen rat finit gen model integr possess quant group symmetry supersymmet model also possess un form dual
sup matrix sup gen linear schrodinger equ shown ar symmetry reduc ordin self dual yang mil equ supergaug group
chern simon the topolog quant field the provid field theoret framework study knot link three dimend rar exampl quant field the exact explicit solv expect valu wilson link op yield class link inv simplest fam jon polynom inv pow jon inv sensit chir knot least upto cross numb unlik jon blind chir howev inv stil good enough distinct class knot cal mut link inv altern obtain dimend vertex model matrix model particul limit spect paramet provid repres braid group turn us construct link inv exploit theorem lick wallac also kirby fen rourk rel three manifold surgery fram link link inv also us construct three manifold inv
construct pair spect paramet ellipt calogero mos hamilton system assocy finit dimend algebra class exceiv typ spect paramet eq three half period ellipt curv result class algebra reduc pair without spect paramet known prevy calogero mos system inv weyl group assocy untw affin algebr simply lac algebra introduc integr system nat assocy twist affin algebra construct op spect paramet exceiv cas
ellipt calogero mos hamilton pair assocy gen simpl algebr shown scal affin tod hamilton pair limit consist tak ellipt modul calogero mos coupl infin keep fix combin delt expon delt crit scal limit ar delt eq coxet numb dual coxet numb untw twist calogero mos system respect limit consist tod system affin algebra limit untw twist calogero mos system delt less crit valu zero consist ordin tod system delt consist trigonomet calogero mos system algebra respect
der expand loop effect act consid first term expand effect act const electromagnet field explicit express next term contain der field strength exact magnitud field strength obtain gen result ferm scal electrodynam pres cas pur elect pur magnet extern field consid detail feynm rul perturb expand loop effect act numb der develop
gen gaug inv margin deform four dimend abel typ topolog field the study shown deform quant field the topolog observ comput addit us link numb smoo intersect ind immers surfac rel eul chern charact class norm bundl und spacetim manifold canon quant the coupl dynam partic string sourc carry hamilton form explicit solv schroedinger equ obtain wavefunct carry dimend unit repres partic string exchang holonom topolog string string intersect holonom giv adiab limit worldsheet eul numb also carry mult dimend project repres derham complex und spat manifold defin gen pres mot group euclid spac arbit manifold pot phys apply topolog field the dual model effect vortex strings discuss
point ambigu regul act second der seem hap multipl standard model el partic
construct irreduc repres extend affin algebr typ coordin quant tor explicit giv formula us vertex op simil found the infinit rank affin algebr infty
dimend is model univers class the whos mass loop tetrahedr diagram unknown analys nat contrast singl scal dimend tetrahedr reduc spec valu expon converg polylogarithm combin dispers rel integ rel find pslq find alph alph thet thet alph arcsin frac empir rel check digit precid ready yield digit transform expon converg akin study math appear dimend result entail polylogarithm lad begin class formul sqrt man dimend result build sqrt
bundl chir block modul spac punct rat curv construct isomorph impl act out automorph und affin algebr bundl isomorph respect knizhnik zamolodchikov connect finit ord prim field fix point isomorph endomorph cas bundl chir block typ reduc vect bundl conject trac endomorph pres propos rel gen verlind formul result apply conform field the bas simply connect group class bound condit the
shown con dougla schwarz compact atrix the noncommut tor prov compact morit equ tor phys equ stat consid gen class dual conject dimend tor
main cont tre conceiv nonperturb lagrang explain extend construct dimend model incorp togeth high dimend the construct method therm entrop property hitherto restrict situ class horizon kil vect gen nonperturb aspect modul loc und algebra less geomet mod think also giv ris interest quest renorm deformatiom high spin field particul gaug the
follow rec propos integr the high dimend bas zero curv lorentz inv submodel princip chir model dimend found infinit loc conserv cur explicit giv cas construct work algebr dimend giv explicit also com apply supersymmet chir model
random driv navy stok equ without press dimend spac consid model strong turb compress fluid der clos equ veloc grady prob dens funct find asymptot funct cas grady veloc field burg turb provid num solv dimend cas apply result veloc diff prob dens funct discuss
form calogero mos model bas root system weyl group pres gen construct pair appl model bas simply lac algebra giv typ cal root minim root typ pair mat us construct bear resembl adjoint repres assocy algebr ex model contain el assocy zero weight correspond cart subalgebr root typ provid simpl method construct sufficy many numb conserv quant model includ bas whos integr unsolv problem twenty year minim typ provid un describ known exampl calogero mos pair cas root typ minim typ form work four cho pot rat trigonomet hyperbol ellipt
gen rec propos noncommut adhm construct cas gamm equ instanton gamm klein group show certain form inhomog adhm equ describ instanton noncommut deform klein orbifold gamm discuss rel nakajim describ instanton spac particul obtain noncommut interpret minim resolv klein singul
complet class supersymmet config bran start oht townsend novel config consid bran mov rel anoth config obtain start coincid bran lorentz transform preserv supersymmetry complet class supersymmet config involv bran interpret group theoret also pres part result supersymmet config involv arbit numb bran show config correspond cayley plan eight dimend nul rot rel remain dimend gen config preserv supersymmetry fract poss restrict plan certain subset cayley grassman discuss exampl fract involv arbit numb bran wel assocy geometry
evalu charact irreduc repres supersymmet extend virasoro algebr der resolv admiss repres also resolv term twist mass verm mod analys charact behav automorph algebr whos sign part spect flow transform poss express charact term thet funct determin behavio spect flow also der id express every charact linear combin spect flow transform charact id involv finit numb charact cas unit repres convers find integr repres admiss charact conto integr admiss charact
provid detail argu der property gen form fact origin propos auth weisz twenty year sol bas assum minim analys valid reduc form property constitut consist equ allow explicit evalu partic form fact scat matrix known equ giv ris matrix rieman hilbert problem exploit shel beth ansatz propos gen formul form fact numb partic sin gordon model alia mass thir model exempl gen solv sev op carry consist check solv three partic form fact thir model perturb the thu confirm gen form
pres last miss detail algorithm class reflex polyhedr arbit dimend also pres result apply algorithm cas three dimend reflex polyhedr polyhedr giv ris surfac embed tor vary contain oth subpolyhedr polyhedr form singl connect defin polyhedr connect contain
quant dynam check matrix spect paramet construct fus proc spin check matrix connect algebr satisfy condit transl inv
nearest neighb point correl funct inv inhomog eight vertex model thermodynam limit comput us fre field repres
consid quant dynam system whos degr freedom describ tim mat plan limit infty exampl gaug theoir atrix the strings stat inv clos strings model trac produc mat discov inv opert act op clos string stat form remark algebr cal heterix algebr simplest spec cas degr freedom extend virasoro algebr infinit dimend gen linear algebr furtherm op act clos string stat form quoty algebr heterix algebr cal quoty algebr cyclix algebr express hamilton gaug field the lik adjoint mat field dimend reduc pur model el algebr fin apply cyclix algebr est isomorph certain plan matrix model quant spin chain system thu obtain matrix model solv plan limit matrix model assocy is model model model satisfy dol grady condit chir pot model thu cyclix algebr describ dynam symmetry quant spin chain system larg gaug field the atrix the strings
construct diagon solv bound yang baxt equ correspond dimend field the quant affin symmetry half lin requir bound unit bound cross symmetry us find overal scal fact lead consist reflect mat us bound bootstrap equ also comput reflect fact scal bound stat breath breath expect ident funda quant partic affin tod field the theref obtain conject affin tod reflect fact comp fact known class result discuss dual property connect particul bound condit
point spat inform encod linear op infinit dimend mat involv describ eith continu discret certain fuzzy spac tim structures argu fuzzy spac structure relev planck scal poss fuzzy spac tim structure rel subtl feat infinit dimend mat analog finit dimend exampl slight weak vert self adjoint symmetry slight weak vert unit isometry rel also spec pres horizon lead mer isomet rath unit tim evolv
many phys system lik supersymmet yang mil the form quant matrix model discuss apply beth ansatz exact solv supersymmet quant matrix model larg limit model construct dimend hubbard model illust
pres survey calibr geometry ar study loc singul structure supersymmet fivebr the particul at geometry plan eight dimend pres result wel many detail comput also analys poss gen self dual geometry afford
consid compact heterot string surfac spin connect embed gaug group dual pict typ string the calab threefold turn ar also dual heterot compact point lik instanton necess singul singul allow ramond ramond modul split distinct compon contain dual heterot tang bundl anoth compon contain point lik instanton pract apply der result heterot string compact tang bundl singul acquir nonperturb enh gaug symmetry fash typ string singul surfac philosoph level discuss appear nat heterot string compact us object der categ coh sheav necess prop extend not dual heterot string surfac
construct gaug the pair dimend noncommut tor lat com intim rel noncommut geometry assocy lat vertex op algebr noncommut tor show trunt tachyon subalgebr nat isomorph class twist mod repres quant deform algebr funct tor construct correspond ev real spect tripl determin morit equ class us string dual argu construct yield simpl proof morit equ dimend noncommut tor giv nat phys interpret term target spac dual group toroid compact string the class automorph twist mod construct gen gaug the inv automorph group comput boson ferm act assocy gaug the show explicit dual symmet dual inv gaug the manifest cov contain high loc interact show also admit sort partic antipartic dual en construct instanton field config dimend dual symmet shel project field the shown coincid standard abel yang mil gaug the minim coupl mass dirac ferm field
start work hypothes phys correspond mathem describ mean discret conceiv planck scal many problem fac enterpr find discret protoform build block ordin continu phys mathem bas approach cal cellul network consist cel nod interact bond fig el interact accord certain loc geomet dynam network liv graph hent subst amount investig devot develop vary vert discret funct analys geometry almost random web anoth import top address suit conceiv intrins fract dimend er structures kind cours investig mak com concern diff rel approach quant grav spin network framework perhap said certain part program seem real idea sketch smolin tim introduc
loop finit temp effect pot extern electromagnet field obtain us worldlin method gen structure temp depend part effect act arbit extern inhomog magnet field est der effect act spin scal stat magnet background der
result symmet loop mass tetrahedron dimend found us lat algorithm pslq giv gen formul involv distinct mass proof dev though account der rath cert ansatz found pslq satisfy easy der pair part diff equ result simil schl afl formul volum rectangul hyperbol tetrahedron rev novel connect loop diagram loop box show reduc common bas volum id tetrahedr correspond loop massless triangl diagram id tetrahedr also obtain evalu volum compl hyperbol knot cas knot posit hent imply field the eas id reduc correl lik appear counterterm volum knot relev numb cont mult loop diagram evalu loop numb goe infin obtain hyperbol volum simpl loop
construct gen calogero sutherland mos system appropry reduc model involv many unit mat result system consist partic circ intern degr freedom coupl mod invers squ pot coupl involv inv ant ferromagnet interact intern degr freedom system shown integr spectr wavefunct quant vert der
consid hungry volterr hierarchy view point mult boson hierarchy construct hungry volterr equ cklund transform ordin on cal fract also study rel discret tim hungry volterr equ matrix model point view study reduc discret tim tod lat discret tim hungry volterr equ
bogomol vort abel hig model study compact rieman surfac gen are volum modul spac comput found depend detail shap surfac volum us find thermodynam partit funct shown thermodynam property depend gen rieman surfac
bradlow limit modul spac bogomol vort compact rieman surfac gen determin hler form volum modul spac comput result comp correspond result prevy obtain gen vortex modul spac
rel soliton solv euclid sin gordon equ lorentz black hol jackiw teitelboim dilaton grav investig emphas import rol play dilaton determin black hol geometry show soliton solv us construct sin gordon coordin black hol mass construct transform standard schwarzchild lik coordin find explicit clos form solv dilaton equ mot soliton coordin find rel soliton paramet black hol mass remark black hol mass neg arbit soliton paramet soliton cas coordin shown cov smooth reg contain whol intery black hol wel finit neighbo outsid horizon hamilton analys perform slic approach soliton coordin intery shown bound contribut intery fin spec sin gordon soliton origin black hol stat mech
dirac conto repres extend parabos paraferm system deform algebr techn analys repres act parapartic annihil op equ deform differenty encod stat parapartic paraferm cas der domain degr polynom
eigenst parabos paraferm cre op construct dirac conto repres parabos eigenst correspond dual vect parabos coh stat ord conserv charg parabos cre op eigenst also construct conto form assocy resolv un obtain
show transf matrix op spin chain diagon bound field symmetry wel dual symmetry interchang exploit symmetry comput exact bound mat regim real
construct trigonomet solv yang baxt equ us fuss cat algebra mult col vert temperley lieb algebr rec introduc bisch jon lead dimend integr lat model describ dens gas col loop
writ vert talk susy pres geomet charact allow near horizon geometry supersymmet bran foc prim bran result bran bran also pres exampl discuss
analys perturb sery expand vacu expect valu wilson loop chern simon gaug the temp gaug analys emerg not kernel vassiliev inv kernel vassiliev inv ord knot inv sint depend regul knot project chos diff vassiliev inv term van knot singul cross conject vassiliev inv reconstruct kernel pres gen form kernel vassiliev inv describ reconstruct ful primit vassiliev inv ord three four ord three recov known combin express inv ord four pres combin express primit vassiliev inv pres ord
restrict affin tod field the half lin impos certain bound condit result the possess spectr soliton breath affin tod the whol lin class solv describ reflect partic bound obtain whol lin kind method mir im depend bound condit chos mir must plac eith front behind bound observ incom soliton convert outgo antisoliton reflect neuman bound condit allow addit solv interpret bound excit bound breath tod the cont most bound condit study among integr bound condit class corrig apply work study vacu solv real coupl tod the half lin perform semiclass calc support rec conject soliton reflect mat gandenberg
finit spin chain bound study der transf matrix diff equ discov cherednik construct eigenst vertex op approach point eigenst magnet field symmetry cal turn symmetry mak symmetry calc spont magnet thermodynam limit rough twic larg half infinit spin chain
verm mod superconfom algebra singul vect spac dimend gre follow method develop virasoro algebr kent introduc conceiv adapt ord superconform algebra prov sev gen result ord kernel assocy adapt ord show siz ord kernel imply up limit dimend singul vect spac apply method topolog algebr obtain maxim dimend singul vect spac topolog verm mod depend typ verm mod typ singul vect consequ prov conject gato river rosado poss ex typ topolog singul vect chir verm mod complet verm mod interest found dimend spac singul vect level fin us topolog twist spect flow also obtain maxim dimend singul vect spac neveu schwarz algebr ramond algebr
form not par period spin chain within quant invers scat method also propos express eigenvalu charg conjug op discret symmetry help class lying stat crit regim giv direct comput matrix
expl rel hopf algebr assocy renorm hopf algebr assocy comput transvers index the foly
rec observ auth supersymmet quant mech susyqm admit form term boson degr freedom construct cal minim boson susyqm appear context integr system dynam symmetry show minim boson susyqm obtain wit susyqm apply nonloc unit transform subsequ reduc eigenspac tot reflect op transform depend par op deform heisenberg algebr reflect intim rel paraboson paraferm emerg nat shown minim boson susyqm also understood supersymmet ferm system interpret boson construct gen cas supersymmetry dimend spec unit transform diagon hamilton op mass fre dirac the result hamilton squ root lik foldy wouthuys cas linear spat der subsequ reduc field compon giv ris linear diff equ reflect whos squ mass klein gordon equ massless limit becom self dual weyl equ linear diff equ reflect admit gen high dimend consist coupl gaug field boson susyqm also gen apply nonloc unit transform dirac field background nonlinear scal field kink config
symmetry espec rel transform reflect equ compon system analys class solv eight sev vertex typ mat giv solv obtain correspond standard mat transform fre ferm model bound mat property fre ferm typ matrix symmetry baxt typ correspond form matrix baxt typ pres hamilton op spin system connect solv particul bound hamilton sev vertex model obtain gen sklyanin form
construct solv chir thir model framework algebra quant field the find posit temp ferm solv coupl const lambd sqrt
solv thir model construct framework algebra shown posit temp ferm solv coupl const lambd sqrt otherw solv anyon diff anyon uncount many liv orthogon spac whol hilbert spac becom sep sect diff urgleichung hold feat certain seen pow expand lambd moreov stat paramet tied coupl const clear expand doom fail nev rev tru structure the bas model quest poss decid wheth ferm boson funda sint dress ferm construct eith bar ferm direct cur algebr
show four der term effect act three dimend yang mil the determin supersymmetry term receiv perturb perturb correct us techn constrain effect act abl determin exact form eight ferm term supersymmet complet term includ instanton correct consequ argu integr eul dens monopol modul spac yang mil determin renorm theorem valu
class rel spin partic model rev quant paramet already class level spec paramet valu emerg requir maxim class glob continu symmetry requir apply rel partic degr freedom giv ris supersymmet quant mech coupl class rel superpartic gaug field ar quant dynam supersymmetry consist supersymmetry appear spec valu coupl const charact interact system boson ferm disappear fre cas poss relev phenomenon high temp superconduc spec
der explicit express correl grassman wel modul spac holomorph map fix degr sphere grassman correl obtain grassmannain first step gen schubert formul self intersect intersect numb modul spac giv explicit clos formula intersect numb found gen altern fibonacc numb pel numb gen random walk partic lin absorb barry intersect numb form wel org pattern
investig linear combin charact minim virasoro model repres produc sev bas block analys bas consid asymptot behavio charact quas class limit particul introduc not second effect cent charg find poss cas fact occ bas gauss jacob watson id exploit result est vary typ id diff charact particul pres sev id gen rog ramanud id apply quas partic repres modul inv partit funct sup conform the conform model bound brief discuss
study fibr target manifold string the provid many insight dual among the ev tool build dual sint work strominger zaslow calab cas the compact joyc manifold fact construct gen kum construct tor smallbb tors fre structure suggest nat fibr smallbb smallbb surfac govern loc pict fibr rol dual string the study intend work achary pres work explain understand glob topolog detail term bundl orbifold ess background provid giv gen discuss fibr gen exceiv fib monodrom bas orbifold bas obtain step routin understand fibr illust exampl turn anoth kind fibr joyc manifold nam fibr calab threefolds construct borce voisin fibr ar fre nat work joyc understand glob structure fibr play rol string the dual maj issu pursuit
artic attempt mad pres rec conceiv comput develop manifest wel establih phys principl vehic convert quant algebra aspect loc quant phys class geomet structures modul the tomit nam laur ded shown togeth collab first tim sufficy gen phys goe einstein caus lin research rec gain moment real structural conceiv innov pow sect also prom comput road nonperturb sect picturesqu speak ent subject extrem opposit noncommut sid
study dirac pres extern chir scal pseudoscal pot prep method develop calc adv retard green funct expand around light con first expand feynm diagram explicit perturb sery light con expand express green funct infinit lin integr extern pot part der dirac decompos caus caus contribut caus contribut light con expand clos rel light con expand green funct describ singul behavy dirac term nest lin integr along light con caus contribut hand every ord perturb the smoo funct posit spac
class gen integr hierarchy assocy affin twist untw moody algebra explicit repres loc conserv dens mean singl scal funct deduc funct act partit funct conserv dens fit pot interpret effect act quant system class consist mult compon gen drinfel sokolov dimend affin tod lat hierarchy rel form approach feigin frenkel enriquez soliton equ mkdv typ also discuss result consid simpl calc conserv charg carry soliton solv equ hierarchy import est interpret partic illust calc charg carry constrain soliton rec construct
spin chain bound study construct vacu stat vertex op level mod ellipt algebr compact geomet symmetry model cal turn symmetry simpl express finit siz formul magnet bound half infinit chain deduc apply formul calc spont magnet bound model
antiferromagnet heisenberg chain map flav massless schwinger model thet electromagnet coupl const veloc light schwinger model determin term heisenberg coupl lat spac spin chain system
univers pair root typ minim typ pres calogero mos model bas simply lac root system includ ex without spect paramet work four cho pot rat trigonomet hyperbol ellipt ellipt pot discret symmetry simply lac model origin automorph extend dynkin diagram combin period pot der class calogero mos model known twist simply lac model untw simply lac model kind root typ pair bas long root short root der contain independ coupl const long short root model contain three independ coupl long middl short root model bas long root exhibit feat deserv study
pres brief deform philosoph ind ref apply quant nambu mech partic phys ant sit spac
project onto gaug inv phys stat rec construct arbit constrain system approach requir gaug fix addit degr freedom beyond origin on charact feat avail method quant constrain dynam work context gen class quant mech gaug inv system cas gaug group consid spec comprehend understand correspond phys spectr achiev straightforward man us standard method coh stat group the direct am gen algebra result extend exampl avail lit much subtl del analys imply gaug fix charact modul spac
quant stat equ qshje der equ principl giv ris init condit seen schroedinger equ ex class limit lead depend integr const planck leng solv qshje provid traject repres quant mech unlik bohm the triv act ev bound stat wav guid pres quant pot turn intrins pot energy partic simil rel rest energy nev van
bas aspect rec propos approach quant mech axiom interpret wav funct mad particul quant pot turn intrins pot energy partic simil rel rest energy nev van rel tunnel effect consequ fact conjug moment field real ev class forbid reg quant stat hamilton jacob equ defin ratio real linear independ solv schroedinger equ theref triv loc homeomorph extend real lin consequ moebi symmetry schwarzian der respect prov bas theorem rel request continu spat infin consequ dual schwarzian der ex solv correspond schroedinger equ result conv approach nee schroedinger equ condit consequ axiom interpret wav funct equ principl imply dynam equ nee assum reproduc tunnel effect energy quant
equ principl stat phys system connect coordin transform fre van energy univoc lead quant stat equ qshje traject depend planck leng hid vary ar init condit form manifest dual consequ involv nat legendr transform rec observ rel second ord linear diff equ reflect intrins dual linear independ solv schroedinger equ unlik bohm the triv act ev bound stat axiom interpret wav funct mad tunnel direct consequ quant pot diff us play rol partic self energy qshje defin ratio loc self homeomorph extend real lin import feat condit us form consequ axiom interpret wav funct direct follow bas theorem us geomet glu condit infty imply result imply dynam equ requir assum reproduc tunnel energy quant sev feat form show copenh interpret hid und nat fin stat high dimend quant equ rel extend der
giv integr bound condit affin tod the soliton preserv sens soliton hit bound reflect soliton prevy known integr bound condit forc soliton convert antisoliton upon reflect prov integr bound condit us gen sklyanin form
describ apply method study discret dynanm system problem continu limit evolv spin network found describ smal scal structure quant gen rel extend conject giv background independ form string the explain us equilibr crit phenomen relev problem continu limit the instead continu limit the lik govern equilibr crit phenomen direct percol fact equilibr crit phenomen self org imply poss class limit quant the grav ex without fin tun paramet
determin spectr cur gen op produc expand energy moment tens sup symmet yang mil the regul term addit multiplet stress tens three cur multiplet appear sigm upsilon start spin respect cur gen infinit tow cur multiplet ev spin exhibit univers structure leng spin unit ident paramet rat famy us high spin techn develop rec conform field the comput crit expon sigm upsilon prov ess structure algebr hold arbit coupl argu algebr clos strongly coupl larg limit result determin quant conform algebr the answ sev quest prevy remain op
valu vacu energy scal field dirichlet neum bound condit infinit clylindr surfac found hap opposit sign contrast class work complet zet funct regul scheme apply field regard interest describ electromagnet cas help fig casimir effect pres surfac found raad milton confirm
us theoret consid show forest formul overlap diverg com hopf algebr root tre
form not par period spin chain within quant invers scat method also propos express eigenvalu charg conjug op discret symmetry help class lying stat crit regim giv direct comput matrix
construct fre field resolv unit repres superconform algebr irreduc repres singl fre field spac cohomolog ferm screening op construct evalu cohomolog resolv assocy ferm screening rel repres the pict gravit descend butterf resolv assocy ferm screenings
construct loc zero curv repres linear sigm model homog spac defin spac tim dimend follow rec propos approach integr the dimend high pres sufficy condit ex integr submodel possess infinit numb loc conserv law exampl involv symmet spac group manifold giv model discuss detail
pres stat analys viewpoint whos mathem bas modul the neuman algebra phys consequ suggest way deal interact symmetry hawk unruh therm property poss also extend scheme renorm perturb the interact incorp us fact matrix rel modul inv interact rel incom wedg algebra point view allow many interest comparid standard quant approach shown firm root hist rad chang paradigm aspect becom partic vis quant meas problem word quant field the matrix the tomit takesak modul the
rec shown renorm quant field the org hopf algebr dec root tre whos coproduc ident diverg requir subtract whos antipod achiev autom process lin recurs symbol cod del finit renorm express feynm diagram thu ver repres op produc expand gen chen lemm it integr subset diagram whos forest structure entail un primit subdiverg provid repres hopf algebr undec root tre undec hopf algebr program design process bphz contribut renorm diagram loop consid model renorm schemes simplest model rev not feat subalgebr con moscovic correspond commut part hopf algebr diffeomorph group assign feynm diagram weight remov zet valu counterterm minim subtract scheme dev fast algorithm weight whos squ sum permut fact giv rat counterterm
rel entrop dimend field the study cylind geometry interpret finit temp field the wid cylind provid infr scal allow defin dimensionless rel entrop analog zamolodchikov funct dimend quant thermodynam entrop giv ris anoth monoton dimensionless quant illust monoton theorem exampl rang fre field the interact model solubl thermodynam beth ansatz dimensionless entrop explicit shown monoton exampl analys
system study exclud stat quas partic conform field the spectr employ method bas recurs rel trunt spectr exampl includ gen ferm unit minim model paraferm spinon wess zumino wit model lat exampl pres explicit express finit affin charact spinon decomposit affin charact
construct quant deform integr mot gen mkdv equ giv relev vertex op algebr prov quant ser rel vertex op allow construct resolv deform class integr mot commut famy
reson vert equ found embed react diffus system equ ant sit valu heisenberg model real particul gaug fix condit jackiw teitelboim grav spac tim point dispers chang sign correspond ev horizon soliton solv black hol soliton veloc bound describ evolv hyperboloid nont wind numb cre collid reson stat spec lif tim
fivebr perturb object string the gen dimend conform field the rel divers subject modul spac vect bundl surfac automorph form ellipt gener geometry calab threefolds gen moody algebra
exact charact traject spec tim prop wign phas spac distribut funct espec simpl indee class quant simpl harmon oscil serv underpin field theoret wign funct form introduc scal field the thu reform term distribut field phas spac apply dual transform field the discuss
canon transform unit counterpart rel rat calogero mos system fre construct
loc analys tod field the not novel chir describ conv nonchir form shown canon interact fre field map describ class solv autom guarantee loc quant tod the investig apply method fre field quant giv tod expon op assocy funda weight vect bilinear form chir field satisfy charact quant exchang algebr shown loc lead nont rel among matrix expand coefficy expon op tod expon obtain system extend algebra method develop liouvil the canon commut rel op field equ also examin
discuss dimend sigm model modul spac instanton surfac superconform field the describ near horizon dynam bran system dual string the der prec rel modul typ string compact modul conform field the correspond class hyperkaehl geometry conclud absens background gaug field met instanton modul spac deg exact orbifold symmet produc turn self dual field deform symmet produc manifold diffeomorph hilbert scheme also com mathem apply string dual glob issu deform hyperkaehl manifold
construct gen calogero sutherland mos system appropry reduc class calogero model subset discret symmetry reduc reproduc known vary system includ rec obtain gen spin sutherland model lead gen ellipt model involv spin inv coupl
defin cohomolog high ord brst ant brst op compact simpl algebr assocy high ord laplac introduc correspond supersymmetry algebr sigm analys op stat gen ferm ghost field transform adjoint repres contrast standard cas laplac giv quadr casimir high ord laplac gen giv complet term casimir racah op involv ghost numb op high ord vert hodg decomposit exhibit exampl work detail includ express high ord laplac
form yang mil the term larg limit view class limit gaug inv dynam vary clos rel wilson loop deform quant obtain poisson algebr dynam vary correspond norm ord quant finit valu hbar op comp poisson algebr introduc past weyl ord quant op find us idea clos rel topolog graph the poisson algebra rough speak prec speak ex invert poisson morph
exposit pap describ geometry certain sasak einstein manifold manifold rec becom interest maldacen conject describ near horizon geometry bran con singul
follow diff track ar definit nahm transform valid self dual field dimend tor zero twist tens transform self dual gaug field defin tor zero twist tens preserv property involv
improv expand wid apply curv domain wal nonrel dissip lambd model real scal ord paramet approxim analys describ domain wal second ord wid pres
shown ex shel light con gaug half ferm compon sup vect pot van part superspac flat condit becom linear reduc spac tim dimend gen solv subset equ der remain linear equ writ form analog yang equ albeit superd involv sixteen ferm coordin shown linear part nevertheless solv method simil pow techn prevy develop pur boson self dual yang mil equ four dimend
pap study structure quant cohomolog ring project hypersurfac posit chern class prov theorem suggest mir transform quant cohomolog project calab hypersurfac clos rel ring symmet funct schur polynom result mind propos gen mir transform quant cohomolog hypersurfac neg first chern class construct explicit predict formul three point gromov wit inv cub rat curv also construct project spac resolv modul spac polynom map good correspond term appear gen mir transform
study connect dual topolog field the first kram wanny dual form simpl topolog claim less poinc dual simil form giv high dimend cas form lead simpl tfts bound colo colo class model poisson dual suggest abel gen cas abel group replac quant group amaz form solv problem without comput quant group appear pict independ class mot connect chern simon the appear symplect level also pict drinfeld doubl reshetikhin turaev inv link manifold comput doubl includ tfts suggest nic phenomen high dimend
investig quant conform algebra supersymmet gaug the phenomen occur strong coupl analys us nachtman theorem gen model independ argu result lead introduc novel class conform field the ident clos quant conform algebr conject exact solv strongly coupl larg limit op conform field the study bas property clos conform field the work op produc expand conserv cur multiplet structure un determin cent charg multiplet contain stress tens also cur finit mass op reason ratio diff hand op algebr contain infinit tow conserv cur org pair singlet respect renorm mix mix second multiplet main consequ diff sublead correct clos algebr simpl consid coincid
monopol soliton import topolog aspect lik quant flux wind numb curv target spac naiv discret substitut lat point und manifold incap retain feat prec study problem discret phys matrix model discuss mathem coh discret monopol soliton us fuzzy phys noncommut geometry fuzzy sigm model act sphere fulfil fuzzy belavin polyakov bound also for
ident clos hyperbol manifold whos volum rat rel dedekind zet valu coprim integ giv zet zet manifold whos inv trac field singl complex plac discrimin degr dedekind zet valu zet largest num inv hodgson week manifold astound tim tim tim largest denomin mer also study rat inv singl complex plac cusp manifold compl knot link within beyond hildebrand week cens within cens ident distinct dedekind zeta rat rel volum moreov cens manifold volum reduc pair zet valu mot study feynm diagram find compon cross link cas altern platon link quart quadr field distinct rat rel dedekind zet valu volum feynm orthoschem find correspond link feynm link miss expect beauty drawn dedekind zet inv obtain knot feynm diagram loop ident sext cross posit feynm knot whos rat inv tim cub cross altern knot maxim symmetry result sec num appear hard prov analys
nonlinear schrodinger equ half lin mix bound condit investig brief introduc correspond class bound valu problem exact second quant solv system construct construct bas algebra structure cal follow bound algebr substitut pres bound famili zamolodchikov faddeev algebr funda quant field the property solv est discuss detail rel scat op der haag ruel framework suit gen cas brok transl inv spac
path integr gen casson inv develop rozansky wit investig path integr vary three manifold explicit evalu class topolog observ introduc allow effect inv fin shown dimend reduc the correspond gen topolog sigm model
investig class modul spac bran nonabel calab threefold singul find admit topolog chang transit construct gen form worldvolum field the langu quiv giv proc comput enlarg kahl con modul spac topolog chang transit achiev vary fayet iliopoulo paramet correspond chang linear geomet inv the quoty study method algebra geometry quit surpr structure enlarg kahl con comput tor method us approach giv detail discuss rank exampl
quant teichmul the bas pen coordin map class group punct surfac repres project cas gen three surfac punct work explicit project fact calc giv expon liouvil cent charg
der microscop spect dens dirac op yang mil the coupl ferm funda repres ess techn ingredy exact rewrit dens term integr sup rieman manifold result agr exact ear calc bas random matrix the
consid donaldson wit the four manifold form tim compact three manifold show interest rel four dimend donaldson inv certain topolog inv particul reinterpret result meng taub rel seiberg wit inv reidem miln tors show partit funct reduc casson walk lescop inv expect form ground cas correct consequ cas observ interest subtl standard expect kaluz klein the apply supersymmet gaug the compact circ smal radi
analys form crossov conduc tens hal plateaux funct extern magnet field propos form crossov obtain act symmetry group particul subgroup modul group up half complex conduc plan assum bet funct describ crossov holomorph funct conduc group act also lead select rul allow transit hal plateaux fil fact
consid problem evolv quant field gen curv cauchy surfac class dynam evolv repres canon transform phas spac field the show canon transform gen unit impl fock spac fre quant field flat spacetim dimend gre consid tim evolv fre klein gordon field flat spacetim toroid cauchy surfac start flat init surfac end gen fin surfac assocy bogolubov transform comput correspond unit transform fock spac mean funct evolv quant stat origin envid tomonag schwinger dirac viabl conceiv nevertheless demonst funct evolv quant stat satisfact describ us form algebra quant field the discuss poss imply result canon quant grav
linear deg poisson bracket antibracket real sol grassman vary pres rev bracket three nilpot delt lik diff op first second third ord respect grassman der shown delt lik op togeth grassman nilpot casimir funct bracket form finit dimend superalgebr
artic techn survey giv pres stat axiom quant field the interest fut direct approach outlin top cov univers structure loc algebra observ rel und field sign rel posit moreov phys interpret the discuss emphas problem appear gaug the revid partic conceiv determin symmetry stat superselect structure analys short dist property spec feat rel therm stat problem appear quant field the curv spacetim also brief ment talk giv ringberg sympos quant field the ringberg castl jun
linear poisson bracket antibracket real sol term grassman vary suggest rev bracket correspond sem simpl group three grassman nilpot delt lik diff op first second third ord respect grassman der contrast canon poisson bracket grassman nilpot diff delt op second ord shown delt lik op togeth grassman nilpot casimir funct bracket form finit dimend superalgebr
dimend chern simon term show spont magnet occ context finit dens vacu lowest landau level ful half occupy ferm charg condens shown appear compl ferm ant ferm condens break flav symmetry caus ferm mass gen solv schwinger dyson equ show ferm self energy contribut induc finit ferm dens ferm mass magnet support charg condens the chern simon coefficy kapp kapp gauss constraint kapp magnet field ferm mass simult gen half fil ground stat break symmetry wel lorentz symmetry
correl funct supersymmet gaug the four manifold sometim express term topolog inv show ex superconform fix point gaug the provid nont inform four manifold topolog particul exampl gaug group doublet hypermultiplet der theorem rel class topolog inv eul charact sign rul seiberg wit inv
discuss aspect heterot string effect field the orbifold construct heterot string calc modul depend threshold correct gaug coupl symmet orbifold compact perform calc threshold correct particul class abel symmet decompos orbifold model intern twist real gen coxet automorph defin limit ex stat caus singul modul spac perturb regim gen vacu heterot string proof provid evid expl stringy hig effect furtherm calc modul depend threshold correct target spac inv fre energy decompos orbifold ident hauptmod funct relev congru subgroup requir solv provid mass term gen effect energy the affect induc sypersymmetry break gaugino condens addit discuss loop gaug gravit coupl decompos orbifold compact second part thes loop correct kahl met gen orbifold compact heterot string calc loop string amplitud prepot vect multiplet effect energy heterot string calc decompos toroid compact heterot string method provid solv loop correct prepot vect multiplet heterot string compact tim
perturb prepot ahl met vect multiplet effect energy heterot strings calc direct dimend toroid compact heterot string vacu method provid solv loop correct vect multiplet prepot compact heterot string rank three four model wel compact tim addit complet prevy calc der string amplitud der diff equ third der prepot respect us complex structure modul tor moreov calc loop prepot us modul property compact heterot string exhibit modul group simil appear sect orbifold bas decompos tor lat supersymmet yang mil
rec pap shown respons integr vary unruh temp comput matrix preserv deform form fact approach giv explicit express deform partic formfact vary integr model sin gordon thir model sev perturb minim cfts real coupl affin tod sery uniform pattern found emerg matrix deform form fact express term barn mult period funct
demonst certain virasoro charact linear combin minim minim conform model admit fact form manifest rel sery permit extract quas partic spectr algebra nat resembl feat tod field the spectr poss admit construct term gen cours analys est interrel fact charact rel paraferm model compact boson minim model
settl app paradox pres therm perturb sery inv manifest temp depend induc chern simon term show larg unlik smal transform hent ward id perturb ord preserv instead therm effect gaug field act induc charg ferm mad inv smal larg gaug transform suit regul dirac op determin us pric par anoma result illust concret exampl
start partic quas exact solv system charact intrins algebra structure energy reflect symmetry construct daught body hamilton pres deform calogero model feat hamilton reduc quadr combin gen interact pot contain body term interact forc cent origin quant valu certain cohomolog paramet quas exact solv multipl stat algebra sect energy reflect symmetry par system preserv
show modul spac littl string the compact symmetry twist eq modul spac instanton commut modul spac instanton commut obtain littl string the bran singul twist larg class gaug the susy susy limit cas the hent modul spac gaug the read modul spac instanton commut tor study phas transit the act dual pur mathem sid giv predict modul spac instanton commut
impl supersymmetry transform hilbert spac op discuss framework supersymmet dynam system shown stat admit impl pur supersymmet ground stat mixt el excit thereof faith stat stat nev supersymmet
construct matrix repres compact membr analys embed complex tor bran config giv ris bergm quant gaug field dual tor almost ant self dual field strength correspond princip bundl shown triv van instanton numb first chern class correspond homolog class membr embed origin tor cours investig show propos quant scheme nat provid assocy star produc spac funct surfac giv explicit coordin inv express produc turn us quant sens deform quant symplect manifold dimend
show algebr morph yang finit algebra correspond nic illust framework linear schrodinger hierarchy purpos giv explicit real yang gen term deform oscil
univers pair root typ spect paramet independ coupl const twist simply lac calogero mos model construct togeth pair simply lac model untw simply lac model pres prevy pap complet der univers pair calogero mos model bas root system twist model bas nroot system typ pot term independ coupl const ad without destroy integr cal extend twist model pair twist model pres exceiv model bas short root pair twist model novel feat der vary funct twist untw appear pair ellipt pot spect paramet provid origin spect paramet also nat explain pair spect paramet twist untw hyperbol trigonomet rat pot model obtain deg limit ellipt pot model
structure heterot string target spac compact study us form noncommut geometry assocy lat vertex op algebra spect tripl noncommut spacetim construct us show intrins gaug field degr freedom disappear energy sect spacetim quant geometry thereby determin much ordin superst target spac set abel gaug the class spacetim ar the effect target spac
rel vacu eigenvalu op spect determin dimend schroedinger op homog pot rec conject dorey tateo spec valu virasoro vacu paramet prov hold suit mod schroedinger op valu
extend prevy introduc construct modul method nonperturb particul relev conceiv quant loc intersect algebra vers class loc support property test funct explain detail wedg algebra construct rig form aspect doubl con algebra fact the determin wel known shel cross symmetry matrix formfact cyc rel the intim rel property quant loc partic pol fre gen wedg algebra gen shel foury transform turn fulfil zamolodchikov faddeev algebr wedg algebra contain cross symmetry inform doubl con algebra rev partic cont field modul the assocy doubl con algebr us chir conform quant field the algebra vert light algebra
abstract form quant dynam pres gen quant constraint develop construct proc relev project op onto phys hilbert spac obtain singl common integr proc origin lagrang multiply vary complet independ gen nat constraint assocy lat limit form demonst expand constraint op contribut second ord lat spac necess us first ord expand nee dynam op contribut among vary poss coh stat path integr us illust complet funct repres abstract quant proc
therm stat chir the rec investig borch yngvason ex rich group hid symmetry show lead rad convers hawk unruh observ follow sens algebra commut algebr assocy heat bath therm chir system us reprocess therm system ground stat system larg algebr larg loc spac tim hap origin system appear kind gen unruh restrict ground stat sytem therm commut transmut new cre virgin spac tim reg behind horizon rel conceiv chir conform cor poss blow lat suggest interest idea loc degr freedom poss repercuss defin quant entrop loc mat cont loc quant phys
review develop past twenty year bas deform philosoph phys the deal el partic compos singleton ant sit spac tim study start kinem aspect espec massless partic extend begin field the composit el partic rel conform field the includ rec develop
consid pair deg compat antibracket satisfy gen axiom impos triplect quant gaug the show act encod group structure antibracket rel left right inv vect field group standard triplect quant axiom correspond abel group
wel known three manifold obtain surgery fram link lick gav el proof ex three manifold inv wit us fram link describ manifold form bracket polynom temperley lieb algebr kaul determin three manifold inv link polynom chern simon the lick formul inv involv comput bracket polynom sev cabl link describ easy obtain bracket polynom cabl us repres the composit braid chern simon the prov cabl correspond tak tens produc funda repres en ver app distinct three manifold inv equ spec rel polynom vary
rec discov form und renorm the hopf algebr root tre allow gen chen lemm gen form describ chang scal green funct hent rel op produc expand hand hand gen goe gen ordin fact tre fact vary id tre fact der clar rel con moscovic weight quant field the
gorsky pres explicit construct whitham deform seiberg wit curv caln susy yang mil the extend result class gaug group cas spect curv affin tod tod system construct us fract pow superpot charact curv also consid plan integr topolog twist the four dimend manifold langu explicit construct whitham deform integr hierarchy typ hid behind
study strong coupl limit flav massless schwinger model lat us stag ferm hamilton approach lat gaug the us correspond lying stat flav strongly coupl lat schwinger model antiferromagnet heisenberg chain est prevy pap explicit comput mass gap excit term vacu expect valu pow heisenberg hamilton spin spin correl funct find satisfact agr result continu the already second ord strong coupl expand show pattern symmetry break continu the wel reproduc lat the indee lat the isoscal isovect chir condens zero every ord strong coupl expand addit find chir condens zero also lat rel lat model ax anoma continu the comput spin spin correl heisenberg model pertin calc mass spectr obtain explicit construct lowest lying stat finit siz heisenberg hamilton chain
consid gen funct prepot gromov wit inv rat ellipt surfac apply loc mir principl calc prepot prov certain recurs rel holomorph anoma equ gen propos holomorph anoma equ gener apply determin high gen gromov wit inv also stat surfac gen ottsch formul hilbert scheme point surfac find prec agr result propos rec mad gopakum vaf
vary fluid mech system enjoy hid high dimend dynam poinc symmetry ar ow desc nambu goto act also reason equ transform diff model interconnect discuss pap
thes strong coupl expand investig multiflav lat schwinger model hamilton form us stag ferm particul interest analys map gaug the onto quant spin antiferromagnet heisenberg chain explot map chir symmetry break pattern study spectr comput extrapol zero lat spac result comp fav qualit quantit weak coupl study gaug model continu
introduc suit adapt ord twist superconform algebr mix bound condit ferm field show ord kernel complet verm mod el ord kernel clos verm mod theref spac singul vect dimend complet verm mod whilst clos verm mod dimend giv singul vect level complet verm mod clos verm mod also giv explicit exampl deg cas dimend singul vect spac complet verm mod gen express conject relev term primit singul vect coefficy respect ord kernel express allow ident deg cas wel clos singul vect also lead discovery subsingul vect twist superconform algebr explicit exampl subsingul vect giv level fin multiply rul singul vect op der us ord kernel coefficy set bas analys twist embed diagram
exclud principl maldacen strominger seen follow deform heisenberg algebra assocy chir ring orbifold cfts deform algebra rel quant group root un interpret algebra spac tim field cre annihil op also propos spac tim origin stringy exclud principl tim spac tim assocy dimend supergrav the acquir quant effect tak account commut structure giv tim remark imply finit effect capt quant group imply prop framework the quest giv grav commut spacetim deform field oscil interest consequ framework holograph interpret produc structure spac unit repres compact quant group root un
pres detail geomet method assocy superalgebr larg class boson supergrav vacu typ correspond el bran the typ string the
introduc gen method transform equ mot follow jevick sakit hamilton bound condit bound valu problem dimend quant mech particul cas dimend chain interact hermit mat correspond larg bound valu problem map linear fredholm equ hilbert schmidt typ kernel equ kernel spec cas second ord diff op allow recov prevy known explicit solv matrix eigenvalu gen cas distribut eigenvalu form der sery saddl point approxim crit behavio system includ prevy observ kosterlitz thouless transit interpret term stat point particul show prevy conject infinit sery lead crit point expand unst stat point consequ real
analys integr property model defin symmet spac dimend us rec propos approach integr the dimend point ingredy the possess infinit numb loc conserv law discuss class model property propos dimend rel inv field the possess toroid soliton solv carry unit topolog charg giv hopf construct act guid requir energy stat config scal inv solv construct exact model possess infinit numb loc conserv cur method also apply skyrm faddeev model integr submodel propos
exposit diff definit approach quant stat giv emphas dimend situ permut stat scat stat exclud stat analys calogero model matrix model spin chain model constitut spec real
clu best understand nat symmetry break mech rev pap revid standard gaug transform property yang mil field accord group approach quant scheme en gaug group coordin acquir dynam cont outsid nul mass shel correspond extr intern field degr freedom transfer vect pot conform mass vect boson
describ pres antisymmet tens tors world sheet act string the rend siz target spac gaug inv quant gen symmetry moment wind exchang whol cruc point transform poss alway lowest eigenvalu hamilton moment mod interpret framework noncommut geometry algebra tak plac point spac spect act principl eigenvalu dirac op funda object the construct quant observ pres many energy eigenvalu dirac op hent hamilton alway interpret target spac string the effect uncompact
pres thet funct repres twist charact rat superconform field the discuss jacob form lik funct property charact fix cent charg act finit heisenberg group modul transform
summ rec result construct pair spect paramet twist untw ellipt calogero mos system assocy arbit simpl algebra scal limit tod system rol seiberg wit the extend part work pres paramet spect curv ellipt spin calogero mos system assocy
describ loc mir symmetry mathem point view mak sev model calc us mir principl loc result agr model comput solv picard fuch diff equ construct form loc geometry near fano surfac within calab manifold interpret gromov wit typ numb enum point view also describ geometry singul surfac show loc inv singul surfac agr smoo cas occ complet intersect
trac anoma extern grav three term crit squ weyl tens eul dens coefficy prop norm cal lat ambigu defin addit const consid unit posit property induc act allow show tot flow eq theref ambigu consist remov id pict emerg clar sev long stand issu interplay unit renorm imply flux renorm group irrevers monoton decreas funct interpol appropry valu nat provid tot flow express perturb inv scheme independ are graph bet funct fix point test predict four loop ord perturb the supersymmet agr also abs interact fix point the argu posit also discuss
form system beth ansatz approach comput bound stat breath mat integr quant spin chain approach calc breath bound matrix op spin chain diagon bound field also comput soliton bound matrix crit regim
summ rec result resolv intim rel problem phys mathem first deal resolv perturb energy dynam certain supersymmet yang mil the cont the mass hypermultiplet adjoint repres arbit gaug algebr second deal construct pair spect paramet certain class mech calogero mos integr system assocy arbit algebr review solv problem wel interrel
poss triv repres gaug group wavefunct gaug inv quant field the lead gen mass intermedy vect tens boson mass paramet show cent charg algebr constraint becom second class nat gaug group coordin acquir dynam outsid nul mass shel provid longitudin field degr freedom massless boson nee form mass boson hig mech could provid clu best understand symmetry break mech un field the un quant massless mass abel yang mil linear grav abel form gaug field the ful develop approach cohomolog origin mass point
investig relev eisenstein sery repres certain inv string the amplitud receiv correct stat stand map class dual dual group respect us inv mass formula construct inv modul funct symmet spac backslash compact typ maxim compact subgroup gen standard holomorph eisenstein sery ar harmon analys funda domain poinc up half plan comp asymptot eigenvalu eisenstein sery second ord diff op quant ar loop string amplitud obtain manifest dual inv repres lat conject perturb dual inv extend analys result perturb effect includ coupl toroid compact the dimend respect
ex intim rel gen stat supersymmetry est observ hid supersymmet structure pur paraboson system structure charact gen nonlinear superalgebr nonlinear supersymmetry paraboson system real turn mod appropry us supersymmet quant mech rel nonlinear paraboson supersymmetry calogero lik model exchang interact spin chain model invers squ interact point
review develop rec propos approach study integr the dimend bas ide consist gen zero curv repres dimend integr model spac tim dimend introduc form connect method us study sev the phys interest lik self dual yang mil the bogomolny equ linear sigm model skyrm typ model loc vert gen zero curv involv algebr repres lead numb conserv law eq dimend repres discuss condit giv the satisfy ord assocy zero curv admit infinit dimend reduc repres also pres the abstract set spac loop giv deep understand simpl form integr dimend
infinit dimend gen quant mech extend object nam quant field the extend object pres paradigm exampl study pap euclid scal field lambd interact four spacetim dimend the found finit virt partic intermedy stat charact fuzzy partic instead ordin pointlik partic caus lorentz inv unit ver four ord coupl const preserv the addit kal lehman spect repres prop discuss
pap elab not fract exclud stat introduc hald dimend conform field the connect univers chir partit funct defin mccoy collab argu gen besid pseudo partic introduc rec guruswamy schoutens nee addit nul quas partic account nul stat quas partic fock spac illust sev exampl model
quant field the extend object employ address hitherto nonrenorm paul interact achiev quant dirac field us infinit dimend gen extend object form ord alph contribut anom magnet mom electron muon ar paul term calc
gen method supersymmet soliton equ hamilton structure est vert supersymmet mod korteweg vri equ giv exampl second hamilton op susy equ gen susy virasoro lik algebr
polysymplect phas spac cov hamilton field the provid cur algebr bracket
vary definit chir observ giv moebi cov dimend the shown equ repres the vacu hilbert spac the study shar gen charact modul inv partit funct although transform property assum first step toward class mad
determin exact quant partic reflect amplitud known vacu affin tod the half lin integr bound condit real singul vacu solv known half class integr bound condit abl fact partic ident analys continu breath solv real vacu solv obtain analys continu stat soliton solv thu obtain partic reflect amplitud correspond breath reflect amplitud turn calc bootstrap soliton reflect mat obtain solv bound yang baxt equ reflect equ study pol structure partic reflect amplitud uncov unexpect rich spectr excit bound stat cre partic bind bound tod the calc reflect amplitud partic reflect excit bound stat abl explain phys strip pol reflect fact eith term bound bound stat gen colem thun mech
discuss promin hopf algebra rec progress quant field the particul consid hopf algebr renorm whos antipod turn conceiv understand subtract proc shal describ sev occ clos rel hopf algebra mathem domain foly rung kutt method it integr multipl zet valu emphas un rol butch group discov study num integr ordin diff equ play
charact linear combin charact admit ferm repres wel fact form consid minim virasoro model consequ vary rog ramanud typ id obtain dilogarithm id produc correspond effect cent charg second effect cent charg der sev way construct gen ferm repres discuss
review apply noncommut geometry system form dual symmetry string the pres spect tripl assocy lat vertex op algebr correspond dirac ramond op construct shown nat incorp target spac discret worldsheet dual isometry noncommut spac target spac dual diffeomorph symmetry shown gaug transform geometry connect noncommut tor matrix the compact also discuss
goal pap analys method angul quant sin gordon model fre ferm point investig model dimend integr field the angul quant method continu analog baxt corn transf matrix method investig canon quant fre mass dirac ferm rindl wedg ident quant repres infinit dimend algebr introduc pap spec fre ferm point construct main ingredy the term repres the algebr follow approach jimbo miw
vary desc dual rel among bran class us topolog the shown desc proc produc typ bran bran antibr bound stat tachyon condens klein project ar nat homomorph group gen bran charg transform gen typ the typ orientifold complet vacu manifold field cont tachyon condens deduc intern desc rel found describ bran orientifold plan topolog defect worldvolum bran antibr pair plan high dimend period property rel shown consequ fact funda bound stat construct hent complet spectr bran charg assocy topolog soliton class four hopf fibr
us fact polychronako lik model obtain freez limit rel spin calogero model calc exact spectr wel partit funct supersymmet polychronako model turn simil supersymmet cas spectr model also eq spac howev deg fact correspond energy level cruc depend valu boson degr freedom ferm degr freedom result partit funct model express novel polynom fin interchang boson ferm degr freedom obtain dual rel among partit funct model
der pow decay asymptot form spin inv wav funct zero mod supercharg reduc dimend supersymmet yang mil the resp matrix model rel supermembr dimend
pres result infinit sery appear feynm diagram calc many simil eul sery includ three dimend sery sery express term zet zet
rec idea gen four dimend gen dimend quant field the term trac anoma extern grav eul dens relev problem quant irrevers ad diverg gaug inv cur extend not eul dens linear conform fact cal pond eul dens not rel trac anoma coefficy univers giv formul express tot flow inv are graph bet funct fix point illust fact detail check predict four loop ord the formul quant irrevers gen ev extend dimend continu although trac anoma extern grav zero dimend show dimend formul predict cont
calogero mos model gen finit reflect group includ model bas crystallograph root system root system finit reflect group dihedr group besid wel known on bas crystallograph root system nam assocy algebra univers pair op gen calogero mos model cho pot construct linear combin reflect op consist condit reduc funct equ coefficy funct reflect op pair four typ funct equ correspond dimend root system root typ minim typ pair der prevy pap giv simplest repres spect paramet depend play import rol pair op bear strong resembl dunkl op pow tool solv quant calogero mos model
brief review progress mad study chern simon gaug the sint rel knot the discov year pres emphas mad analys perturb study the connect the vassiliev inv describ study quant field the three diff gaug fix lead three diff repres vassiliev inv gaug fix lead wel known repres cov landau gaug correspond config spac integr cov light con gaug kontsevich integr progress mad analys third gaug fix cov temp gaug describ detail cas obtain combin express instead integr on vassiliev inv approach bas last gaug fix seem prom obtain ful combin formul collect combin express vassiliev inv ord four obtain approach
infinit dimend gen quant mech extend object nam quant field the extend object employ address hitherto nonrenorm gravit interact follow cosmolog const problem address respons electron weak gravit field linear approxim study ord alph correct magnet gravit mom comput
determin topolog superconform algebr pres wel detail analys singul vect detect root determin addit ident standard verm mod contain label singul vect detect direct root determin show standard verm mod least four diff typ submod regard siz shap also review chir determin formul chir verm mod ad insight fin transf result obtain verm mod singul vect ramond algebr poor study work clar sev misconceiv confus claim appear lit singul vect verm mod submod topolog superconform algebr
pap construct explicit infinit numb hopf stat soliton solv zero hopf topolog charg within rec propos dimend integr rel inv field the integ label famy hopf found produc eq hopf charg provid low bound soliton finit energy hopf construct explicit term toroid coordin shown form link clos vort
renorm group method pow tool reduc evolv equ form term not inv manifold start der exact equ analog wilson equ stat phys quant field the clar perturb method construct inv manifold success init valu evolv equ thereby mean nat understood arbit init tim show integr const unperturb solv constitut nat coordin inv manifold linear op evolv equ jord cel jord cel slight mod necess dimend inv manifold increas perturb equ determin slow mot would integr const unperturb solv inv manifold pres mech proc construct perturb solv hent init valu equ giv mean result und structure reduc method form pres work turn complet univers elucid kuramoto year ind reduc proc evolv equ good correspond renorm proc quant field the count part univers structure reduc elucid kuramoto polchinsk theorem renorm field the apply method interfac dynam kink ant kink soliton soliton interact lat linear op jord cel structure appear
shown modul grad gen twist funct dynam system proport ordin stat twist weak in correspond repres funct inv adjoint act asymptot abel famy automorph twist triv consequ funct ex supersymmet dynam system contrast situ compact spac sup funct occ sup gib funct
infinit sery grassman grassman ev flow equ defin class supersymmet integr hierarchy assocy loop superalgebra flow commut mut commut boson on origin consid defin hierarchy hent provid extr ferm boson symmetry includ built supersymmetry transform correspond loc conserv quant also construct exampl particul cas princip supersymmet hierarchy assocy affin superalgebra ferm simpl root system discuss detail
study yang symmetry multicompon quant nonlinear schrodinger hierarchy framework quant invers scat method giv explicit real yang gen term deform oscil algebr nat occ framework
shown spac tim coordin exhibit typ short dist structures describ linear op continu discret unsharp way lit vary quant grav model spac tim short dist point toward typ unsharp investig property field unsharp coordin find field continu possess finit dens degr freedom simil field lat observ typ unsharp techn apert induc unsharp opt im also typ unsharp tim resolv bandlimit electron sign indee spec cas recov shannon sampl theorem inform the
conject abel project respons confin col us gaug independ definit abel project employ gaug fix provid strong evid abel domin wilson loop integr spec prov gaug pot contribut wilson loop integr prec restrict abel project
topolog ory spacetim capt certain lump triv topolog cal topolog geon topolog analog conv soliton giv describ topolog geon degr freedom rel topolog sep complet the contain met dynam degr freedom form also allow investig process quant topolog chang correspond cre annihil quant geon select rul process der
show singl mod paraferm typ system possess supersymmetry bas symmetry charact funct paraferm rel gen deform oscil daskaloyan supersymmetry real unbrok spont brok phas cas paraboson supersymmetry observ rec auth form assocy superalgebr depend ord paraferm linear nonlinear hamilton list supersymmet paraferm system includ us paraferm finit dimend deform oscil deform paraferm oscil paraferm oscil intern structure
analys highest weight repres ramond algebr show structure rich prevy suggest lit particul show certain verm mod ramond algebr contain deg dimend singul vect spac supersymmet cas ev contain subsingul vect choos suit ord ramond algebr gen comput ord kernel turn dimend complet verm mod dimend clos verm mod dimend ord kernel allow der multiply rul singul vect op lead express deg singul vect us multiply rul study descend singul vect der ramond embed diagram rat model giv explicit exampl singul vect deg singul vect subsingul vect level conject ord kernel coefficy primit singul vect theref ident vect un
vacu pass uniform accel observ ant sit spac tim us op perpetu mobl reg univers valu hawk unruh temp discov symmetry find observ compl wedg shap reg commens loc vacu stat result model independ hold the compat weak not spac tim loc
der gen skyrm faddeev act effect act energy limit result demonst ex mass trig confin col
review aspect integr system discov rec supersymmet gaug the topolog twist vert main top explicit construct whitham deform seiberg wit curv class gaug group apply contact term plan integr topolog twist the connect funct blowup formul topolog twist the
examin dynam induc four dimend bound fiv dimend ant desit spacetim fiv dimend chern simon the gaug group direct produc show giv bound condit compat geometry spacetim asymptot reg induc surfac the model
fracton anyon class equ class obey spec fract stat equ class label fract paramet hausdorff dimend consid approach context fract quant hal effect fqhe conceiv dual class defin tild show fil fact fqhe observ appear class connect equ class modul group quant phas transit fqhe also obtain bet funct defin complex conduc embody class thermodynam also consid fracton const dens stat exact equ stat obtain temp dens limit also prov farey sequ rat numb express term equ class
discuss loc mir symmetry high gen curv spec consid topolog string partit funct high gen curv contain fano surfac within calab main exampl loc cas kodair spent the grav tail loc geometry solv comput partit funct us result gopakum vaf loc mir partit funct rewrit term expand coefficy found integ ver loc calc model many gromov wit predict integr mystery mathem speak asymptot grow degr inv analys suggest mad toward enum interpret follow stat describ gopakum vaf
review artic introduc un fash intermedy step difficult calc infinit dimend algebra quant matrix model op string sect clos string sect phys observ quant matrix model larg limit express el algebra algebra ar quoty algebra larg algebr also discuss property algebra publ elsewh brief review rel wel known algebra lik cuntz algebr wit algebr virasoro algebr also review yang mil the vary energy effect model string the quant grav string model quant spin chain model form quant matrix model study algebra thu help understand common symmetry phys system
comput beth ansatz bulk bound hol scat mat crit quant spin chain bulk matrix coincid soliton matrix tod field the imagin coupl ver result bound matrix us gen ghosh zamolodchikov bound cross rel
consequ eyvind wichman contribut modul the phas spac structure pres ord show pow idea contemp problem select issu algebra holograph wel nonperturb construct approach bas modul structur wedg loc algebra modul includ show idea rec consequ pathbreak work wichman togeth collab bisognano in
self dual chern simon soliton influ quant pot consid singl valu condit arbit integ numb soliton lead quant chern simon coupl const kapp integ strength quant pot show jackiw model correspond first memb hierarchy chern simon gaug nonlinear schr od model admit self dual soliton typ expon loc chern simon soliton bloch electron near hyperbol energy band bound found
thermodynam beth ansatz equ coupl linear integr equ appear frequ solv integr model assocy model supersymmetry rel diff equ among painlev tod hierarchy simplest cas massless limit linear integr equ solv term airy funct known clos form solv thermodynam beth ansatz equ outsid fre class model turn giv spect determin schrodinger equ linear pot
der vert abel stok theorem mak coh stat repres coset spac tim flag spac outlin der are wilson loop yang mil the maxim abel gaug detail exposit giv forthcom artic der perform combin abel stok theorem reform yang mil the perturb deform topolog field the rec propos auth within framework show funda quark confin ev brok part gaug fix brok tim origin are rel geomet phas wilczek holonom abel domin immedy byproduc result magnet monopol play domin rol der
scat scat describ propos bound perturb inv conform field the bulk massless mat form zamolodchikov staircas model us bound vert thermodynam beth ansatz show bound fre energy goe sery integ valu plateaux funct system siz
find solv singul vect decoupl equ point funct particul cas field weight cdot lambd result funct triv singul flag vary nam linear combin hypergeomet funct calc fil confirm fus rul determin gen kapp fract level also analys fus us algebra method gen feigin fuch find agr process clar detail prevy tre fus fract level admiss repres
form dimend yang mil the adjoint mat field larg limit class mech der poisson algebr col inv observ involv adjoint mat field show rig math phy diff quant ord observ produc ess poisson algebr explain less prec pedagog man cruc topolog graph observ und form proof
wel known spit shar property conv partic topolog geon gen viol spin stat theorem hand gen believ quant grav the allow topolog chang us pair cre annihil geon abl recov theorem pap tak altern rout algebra form develop prevy work giv describ topolog geon algebr observ ident quant diff irreduc repres algebr correspond diff kind geon label abel charg magnet flux find us spin stat theorem indee viol spin stat rel ar assum flux superselect assum prov observ loc gen cas phys the fin also show approach fit conv form quant grav
pres strong evid magnet confin demonst loop effect act induc dynam symmetry break thorugh monopol condens could induc dual meissn effect guar confin col abel gaug the result obtain sep topolog degr describ abel monopol dynam degr pot integr dynam degr
op form boson finit temp dens develop tre gen cas anyon stat exact point correl funct satisfy kubo martin schwinger condit explicit construct inv vect chir transform allow introduc chem pot investig exact moment distribut discov anyon condens certain rang stat paramet anoth interest feat occur van persist cur apply gen form solv massless thir model finit temp der charg dens persist cur
develop gen framework quant field the noncommut spac spac quant group symmetry path integr approach obtain express point funct perturb the lead gen feynm diagram braid triv cross demonst pow approach apply the quant sphere find bas diverg diagram the regul
gen calc casimir energy arbit numb dimend massless quant field spherically symmet cav carry common situ includ scal spin field electromagnet field vary bound condit tre car fin result giv analys clos express term barn zet funct direct straightforward num evalu formula perform yield high acc numb principl arbit good precid
conceiv determin linear op infinit dimend spac address us der op zet funct follow sing ev zet funct trac littl play op simpl id op vary thereof show pres commut anoma fact unavoid ev commut remark also almost const op cas dirac typ op simil bas argu lead conclud contradict common lor spit slash slash follow symmetry condit slash spectr turn determin eq sqrt slash simply slash slash slash slash proof fact giv simpl exampl us op harmon oscil spectr fulfil symmetry condit anoma phys relev addit mass term instead chem pot contribut ad dirac op
giv repres ellipt hyperbol homog second ord mong amp equ connect equ equ hydrodynam typ giv scal dispersionless repres matrix dispers repres follow correspond sigm model paramet equ minim surfac mong amp equ loc wel nonloc conserv dens obtain
attempt defin inv almost calab fold count spec lagrang rat homolog spheres homolog class certain weight depend topolog mot gromov wit inv symplect manifold count holomorph curv homolog class ord inv interest eith unchang deform und almost calab structure els transform accord rigid rul period almost calab structure pass topolog determin hypersurfac cohomolog deform und almost calab fold collect spec lagrang homolog spheres chang becom singul thu determin stabl inv deform nee know singul behavio spec lagrang fold wel understood describ kind singul behavio spec lagrang fold der id weight funct unchang transform wel weight funct satisfy id conject inv defin weight independ kahl class chang certain way holomorph form pass real hypersurfac fin consid connect string the argu inv count isol bran play part mir symmetry story calab fold
univers disentangl formul bak campbel hausdorff coh stat perelomov typ ad defin gen oscil algebra giv
prec rel arbit monodrom depend form appear chir wznw symplect form exchang matrix govern correspond poisson bracket est gen ear result rel diagon monodrom exchang mat shown satisfy dynam gen class mod yang baxt equ found admit interpret term poisson groupoid dynam exchang mat right multiply yield class poisson symmetry chir wznw phas spac pres explicit
calogero mos model tod model wel known integr mult partic dynam system bas root system assocy algebra rel typ integr model investig level hamilton pair pair calogero mos model spec repres reflect group correspond algebra lat spec pair tod model hamilton ellipt calogero mos model tend tod model period ellipt funct goe infin provid dynam vary prop shift coupl const scal hand calogero mos pair exampl root typ pair consist tod model limit minim typ pair correspond minim repres algebra tend pair correspond tod model
pap count numb real complex solv beth constraint partic sect model find exact numb exceiv string conject tot numb solv requir complet
construct georg glashow lagrang gaug group break symmetry spont giv depend mass gaug field vacu manifold turn vacu manifold paramet commut quant show monopol solv ex model ind pres stat
shown con dougla schwarz gaug the noncommut tor describ compact the tor const background three form field ind noncommut gaug the gen manifold also us string the discuss framework noncommut quant gaug the poisson manifold us deform quant kontsevich formul star produc giv origin term perturb expand lead renorm quant field the discuss nonperturb path integr form cattaneo feld poss approach construct noncommut quant gaug the poisson manifold aspect class quant noncommut field the also discuss
topolog quant field the us pow tool prob geometry topolog dimend chern simon the exampl field the provid field theoret framework study knot link three dimend rar exampl quant field the exact perturb explicit solv abel chern simon the provid field theoret interpret link self link numb link abel the vacu expect valu wilson link op yield class polynom link inv simplest fam jon polynom inv obtain pow jon pow method complet analys perturb comput knot link inv develop process answ op problem knot the obtain inv un fram link inv three manifold easy construct exploit lick wallac surgery pres three manifold inv norm partit funct chern simon field the ev perturb analys chern simon the rich mathem structure provid field theoret interpret vassiliev knot inv mathem chern simon the find import apply three four dimend quant grav also
exploit property deform coxet el scat mat affin tod field the real coupl const rel dual pair simpl algebra express complet gen discuss govern equ ex bound stat fus rul term deform coxet el twist deform coxet el undeform coxet el est prec rel diff form study solv gen matrix bootstrap equ shown equ fus rul rel diff vert fus rul quant conserv quant result nullvect doubl deform cart lik matrix pres property matrix togeth cal combin bootstrap equ util ord der gen integr repres scat matrix term quant eith dual algebra pres extend cas cas dat particul orbit gen vary coxet el
quant matrix model larg limit ar many phys system lik yang mil the without supersymmetry quant grav string model vary energy effect model string the atrix the quant spin chain model strongly correl electron system lik hubbard model introduc un fash hierachy infinit dimend superalgebra quant matrix model superalgebra pertain op string sect anoth clos string sect phys observ quant matrix model lik hamilton express el superalgebra ind superalgebra describ symmetry quant matrix model pres structure superalgebra lik cart subalgebra root vect id subalgebra gen wel known algebra lik cuntz algebr virasoro algebr toeplitz algebr wit algebr ons algebr lik learnt crit phenomen string the conform symmetry describ virasoro algebr learn quant chromodynamics quant grav condens mat phys symmetry quant matrix model describ superalgebra
detail review rec develop topolog class bran superst the pres begin thorough self contain introduc techn apply topolog the rel class construct the rec real bran tachyon soliton com bound stat high dimend system unst bran describ shown the form nat reproduc known spectr bran system us predict ex stat emphas plac interpret bran conv topolog soliton bran worldvolum mathem form us deduc gaug field cont supersymmet bran also the predict rel vary superst the bran spectr impl dual symmetry nat isomorph group discuss rel standard cohomolog class pres us der explicit formul bran charg string theoret construct the predict also brief describ
respons integr vary unruh temp rec shown comput matrix preserv replic deform form fact approach show replic deform form fact inv thir model found among solv rat typ quant knizhnik zamolodchikov equ gen level show modulo conserv charg solv deform form fact correspond on level zero conject deform form fact noeth cur model
quant the real degr freedom us describ op self adjoint howev exceiv rul infinit dimend hilbert spac op necess self adjoint ev expect valu real instead op mer symmet op diagon consequ describ real degr freedom display form unsharp fuzzy exampl ind typ op could ar describ spac tim string planck scal form unsharp fuzzy long conject prior howev pot problem mer symmet op fact unlik self adjoint op gen unit least straightforwardly show larg class op gen unit wel defin op ev gen entir unit group hilbert spac show mer symmet op addit describ unsharp phys ent indee also play gen symmetry within funda the quant grav
us plan integr moor wit der simpl express donaldson inv sigm tim sigm rieman surfac gen express gen theorem morg szabo gen giv apply result der thadde formula intersect pair modul spac rank stabl bundl rieman surfac der eigenvalu spectr fukay flo cohomolog sigm tim
discov real affin superalgebr exceiv affin superalgebr alph vertex op extend affin algebra dual level auxy level algebr dual rel level construct repres level tens produc level level level mod decompos direct spect flow orbit decomposit giv ris charact id also der extend construct affin level trac property embed alph rel dual pair convers show level repres construct level repres
rec kontsevich found combin formul defin star produc deform quant poisson manifold kontsevich formul reinterpret phys quant correl funct topolog sigm model op strings wel context bran flat background neveu schwarz field correspond kontsevich formul dual algebr der term form bran group manifold particul show formul encod point correl funct wess zumino wit effect the dirichlet bound condit field ent form play import rol der
analys glob anoma el typ strings pres bran glob anoma cancel giv restrict bran topolog restrict mak poss interpret bran charg el the
rec idea modul loc loc quant phys us clar rel shel quant partic phys particul rel shel cross symmetry shel einstein caus among col result nonperturb approach profound rel cross symmetry partic phys hawk unruh lik therm aspect property entrop attach horizon quant mat behind caus horizon aspect hitherto exclud rel kil horizon curv spacetim rath loc aspect minkowsk spac partic phys scop modul framework amaz wid rang provid conceiv bas bootstrap formfact program fact model decomposit the term finit collect unit equ chir conform the plac spec rel posit within common hilbert spac holograph rel high dimend lik scan framework giv spacetim interpret zamolodchikov faddeev algebr explain therm aspect
apply geomet inspir repres epsilon expand point funct diff mass consid explicit result arbit term expand obtain term sin integr construct epsilon expand three point cas also discuss
canon quant abel typ topolog field the coupl extend sourc gen dimend manifold curv lin bundl study sheaf cohomolog us construct appropry topolog extend act topolog flux quant condit term cech cohomolog und spat manifold requir topolog inv quant field the wavefunct found hamilton form shown carry mult dimend repres vary topolog group spac express gen link numb dimend thereby der particul glob aspect mot group pres obtain dimend apply quant exchang stat object vary dimend also discuss
develop graph repres polynom inv unit gaug group find algebra curv correspond hyperkahl quoty linear spac apply method four dimend spac cas der explicit rel deform curv away orbifold limit fayet iliopoulo paramet correspond quoty construct work orbifold limit high dimend exampl
defin sem hopf algebr gen hopf algebr construct supersymmetry algebr adjoint act sem hopf algebr result supersymmetry the quant gaug group quant envelop algebr simpl algebr exampl construct lagrang supersymmetry
method develop construct solv many vary linear diff equ arbit ord us partic sutherland model pair wis invers sin squ interact among partic shown equ fre partic circ appl method many many body problem also illust
analys high temp class limit casimir effect us quant ar nat discuss rel casimir energy defin config disjoint conduc bound arbit shap diff casimir energy giv config config bound infinit apart us path integr techn show rel casimir energy van expon fast temp consist simpl phys argu bas kirchhoff result rel casimir entrop defin obvy analog man tend class limit finit asymptot valu depend geometry bound thu casimir forc disjoint piec bound class limit entrop driv govern dimensionless numb charact geometry cav contribut casimir thermodynam quant individ connect compon bound exhibit logarithm devy temp behavy describ logarithm devy seem ar difficul sep casimir energy thermodynam quant electromagnet self energy connect compon bound wel defin man approach casimir effect impos sharp bound condit fluctu field rath tak consid interact plasm charg carry bound plasm frequ play rol phys cutoff also allow analys devy perfect conduc behavy
us property determin lin bundl famy ellipt bound valu problem explain fock spac funct defin axiom quant field the form model ferm path integr sew axiom the ar algebra past determin dirac op show repres bound gaug group describ lead fock funct describ certain gaug anoma
discuss superalgebr project the typ algebra complet class second rank cent term valu algebr find poss stat com term show among typ susy config correspond sup bran anoth ar nilpot el
study spin spin spin energy energy energy correl is model perturb magnet field comp result high precid montecarlo sim predict diff approxim form fact approach bas exact matrix describ model short dist perturb expand around conform point method giv good result first perform bet dist larg correl leng second prec dist smal correl leng ord improv agr extend perturb analys second ord der const
techn not giv pur geomet understand discret tors analog orbifold wilson lin form tens field pot ord introduc discret tors context describ gerb describ certain typ supergrav tens field pot connect gerb discret tors nat appear describ act orbifold group gerb orbifold wilson lin appear describ act orbifold group gaug bundl result restrict triv gerb word describ discret tors apply eq wel compact nont field strengths also describ specir program rig der analog discret tors many typ tens field pot abl mak spec conject result
consid dimend integr conform inv field the possess dirac spin three scal field interact coupl bilinear term spin expon scal integr property bas affin moody algebr simpl exampl cal conform affin tod the coupl mat field show us boson techn class equ noeth cur topolog cur hold tru quant level lead model lik mech confin spin field insid soliton boson spin show the decoupl sin gordon model fre scal construct soliton solv show interact lead tim delay sin gordon soliton model provid good lab test dual idea context equ sin gordon thir the
prevy pap outlin discret tors understood geomet analog orbifold wilson lin pap shal prov remain detail prec pap describ gerb term object known stack ess sheav categ develop much bas the gerb langu relev technolog describ giv first principl geomet der discret tors word defin equ gerb class equ structures gerb gerb connect prov gen equ structures gerb connect tors group includ orbifold group spec cas triv gerb equ structures furtherm canon ident group
show renorm quant field the spec inst gen mathem proc multiply extract finit valu bas rieman hilbert problem giv loop gamm el complex group gen proc giv evalu gamm perform birkhoff decomposit gamm gamm gamm gamm loop holomorph in out domain rieman sphere gamm infty show us dimend regul bar dat quant field the del loop devy complex dimend el dec butch group obtain us miln moor theorem kreim hopf algebr renorm gen proc del renorm phys the minim substract scheme
soliton tim delay semiclass limit soliton mat calc simply lac affin tod field the phas shift writ bilinear soliton conserv charg result apply soliton affin tod field the produc gen express numb bound stat valu coupl matrix diagon obtain ord ar result vertex op construct princip grad simply lac affin algebra extend prevy construct simply lac twist affin algebra
pres investig bound breath stat sinh gordon model restrict half lin class bound breath pres paramet famy integr bound condit restrict cas bound condit preserv symmetry bulk the energy spectr bound stat comput way first us bootstrap techn subsequ us approxim requir describ spectr agr allow determin rel bound paramet bulk coupl const paramet appear reflect fact der ghosh describ scat sinh gordon partic bound
show conform map simply connect domain analys bound unit disk intim rel dispersionless tod integr hierarchy map determin particul solv hierarchy singl condit known string equ hierarchy loc solv invers pot problem reconstruct domain harmon mom solv known describ grav coupl mat also introduc conceiv funct analys curv
show rec form equ principl imply bas cocyc condit euclid minkowsk spac hold dimend condit dimend sufficy schwarzian equ imply funda high dimend moebi inv turn univoc fix quant vert hamilton jacob equ hold also rel cas obtain tim depend schroedinger equ klein gordon equ dimend show imply mass rel map induc coordin transform connect diff phys system furtherm show minim coupl prescrib theref gaug inv ar quit nat impl fin show antisymmet tens und quant mech shed light nat quant hamilton jacob equ
study deform quant scal abel gaug class fre field stratonovich weyl quant star produc wign funct obtain field oscil vary abel gaug the particul intrigu sint wign funct fact phys part contain constraint effect triv topolog within deform quant form also consid
shown calogero mos model bas root system finit reflect group crystallograph crystallograph cas rat without harmon confin pot trigonomet hyperbol pot simply supersymmet term superpot univers formul supersymmet ground stat wavefunct sint boson part supersymmet model us quant calogero mos model giv univers formul ground stat wavefunct energy determin pur algebra quant pair op conserv quant calogero mos model est
algebr funct deform ant sit spac defin cov root un star structure study detail scal field intrins high energy cutoff ar nat field orbifold tim tim ev finit abel group certain chir sect class sphere hilbert spac squ integr funct discuss analog result found deform sphere
zet funct assocy massless mass dirac op dimend ev manifold without bound rig construct sev mathem subtl involv process stressed int ambigu pres definit assocy ferm funct determin massless cas also unavoid pres situ multiply anoma conveny express term noncommut residu ambigu seen disappear mass cas giv ris phas dirac determin agr rec calc appear mathem lit multiply anoma also agr calc coincid situ explicit nont resum mass sery expand involv zet funct result express term quit simpl formula
chir wznw symplect form omeg chir invert gen cas thereby prec rel arbit monodrom depend form appear omeg chir exchang matrix govern poisson bracket group valu chir field est exchang mat shown satisfy dynam gen class mod yang baxt equ poisson groupoid construct encod equ analog group encod class equ arbit simpl group exchang mat found correspond poss structures admit symmetry
describ spac tim ferm dirac ahl field inhomog diff form provid interest altern dirac spin field pap develop simil conceiv within symplect geometry phas spac rath spac tim symplect dirac ahl field defin class phas spac hamilton system equ infinit famy metaplect spin field spin ordin dirac ahl field equ finit mulitplet dirac spin result interpret framework gaug the form quant mech propos rec intrigu analog found lat ferm problem specy doubl problem quant gen
exhibit rel massless integr quant field the certain third ord ordin diff equ thereby extend rec result connect massless sin gordon model schr od equ form part gen correspond involv rel beth ansatz system third ord diff equ linear integr equ gen spect problem der num check perform dual property discuss simpl vary nonlinear equ suggest candid describ finit volum ground stat energy minim conform field the perturb op check prevy result obtain us thermodynam beth ansatz
rec theoret num develop show analog quant chromodynamics disord system condens mat phys study spect fluctu dirac partic prop finit four dimend pres gaug field construct model combin efetov approach disord system principl chir symmetry gaug field replac stochast whit nois pot gaug field disord effect supersymmet linear sigm model obtain spont break supersymmetry found rig der equ thouless energy connect energy effect the particul nambu jon lasinio model chir perturb the found
wel known chir wznw bloch wav satisfy quadr class exchang algebr imply affin moody algebr correspond cur obtain direct der exchang algebr invert symplect form spac bloch wav giv complet algorithm construct gen fre field real extend class wakimoto real cur algebr
subsingul vect superconform algebra discov exampl giv short afterward semikhatov tipunin claim obtain complet class subsingul vect pap structure verm mod superconform algebr publ surpr explicit exampl subsingul vect known tim class result pres pap includ class subsingul vect bas follow assum auth claim diff typ submod verm mod overlook begin indecompos label singul vect discov month clear typ submod auth claim construct conv singul vect property gen typ submod maxim subsingul vect left outsid not prov assum incorrect fact also affect profound result pres sev publ espec pap equ affin semikhatov embed diagram verm mod semikhatov sirot singul vect semikhatov tipunin last rev vert septemb
algebra structure rel eigenspac calogero sutherland model sutherland model circ investig cherednik op find exact connect simult symmet eigenfunct cherednik op eigenfunct construct monom construct allow writ harmon oscil algebr involv cherednik op yield rais low op model also show connect fre oscil fre partic circ also point subtl diff excit
continu phys import topolog aspect lik instanton thet term ax anoma conv lat discret oft difficul tre aspect pap develop discret quant field the fuzzy manifold us noncommut geometry bas prevy tre instanton chir ferm without ferm doubl fuzzy spac espec fuzzy spheres pres discret repres thet term topolog suscept gaug the der ax anoma fuzzy sphere gaug field act four dimend bound modul instanton numb continu
left right zero mod level wznw model giv ris pair isomorph left right mut commut quant matrix algebra deform paramet ev root un matrix algebra admit id correspond fact algebr finit dimend structure superselect sect diagon wznw model reduc finit dimend problem gaug the typ problem solv us gen form
pap devot gen self contain approach cohomolog field the hler structure
lect discuss beyond symmetry point view review virasoro algebr introduc deform virasoro algebra ellipt algebra algebra appear solv lat model study fre field approach
rec pap saveliev auth show exit shel light con gaug linear part field equ reduc sup vert yang equ solv method inspir on prevy develop self dual yang mil equ four dimend lat analog lat the pres on push writ sup part linear diff equ analog pair belavin zakharov us repres shown pres artic solv gen techn work simil on develop four dimend self dual yang mil the lat seventy
discuss stat perspect rel quant phys
der formul express loc spin field op funda grad model term el monodrom matrix formul quant analog class invers scat transform apply funda spin chain chain numb import exact solv model strongly correl electron supersymmet model model
continu exact integr techn mod dynam equ dimend supersymmet gaug the acklund transform der yang typ sup equ prevy der saveliev auth dimend sup yang mil field equ shel light con gaug shown bas upon particul gaug transform satisfy nonlinear condit ens particul form equ retain yang typ field equ shown prec autom provid solv condit acklund transform simil propos lesnov self dual yang mil four dimend introduc person recollect bir supersymmetry giv
pap study sev issu rel gen superpot induc background ramond ramond flux compact typ string the calab four fold ident soliton bran wrap calibr submanifold calab spac propos gen formul superpot just comp supersymmetry condit supergrav the also suggest geomet interpret supersymmet index dimend effect the term topolog inv calab four fold estim asymptot grow inv black hol entrop fin explicit construct supersymmet vacu typ string the compact calab four fold ramond ramond flux
mot form noncommut produc system op strings bran const nonzero neveu schwarz form defin deform multiply op quasitriangul hopf algebr term matrix com property show noncommut string the produc particul exampl multiply com poss hopf algebra property und the
born infeld nonlinear electrodynam point singul elect magnet charg consid problem interact assocy soliton dyon solv investig cas long rang interact first ord smal field dist soliton obtain gen lorentz forc act dyon consid short rang interact dyon id elect opposit magnet charg investig init approxim consid cas veloc dyon eq mod opposit direct common lin shown assocy field config const ful angul moment independ interdyon dist spee property permit consid bidyon config electromagnet model charg partic spin num investig mov dyon config cas ful elect charg eq electron charg ful angul moment eq electron spin shown cas absolv valu rel elect magnet charg dyon eq fin structure const calc giv bidyon behav nonlinear oscil assocy depend frequ ful energy obtain init approxim limit electrodynam model obtain quick oscil wav packet behav lik mass gravit partic mov high background field discuss poss electrodynam world oscil bidyon partic
nat analys cqft correspond context conform compact cov form obtain addit insid rehr rig algebra holograph connect degr freedom issu turn allow illust subtl import diff beween origin string the bas maldacen conject rehr theorem set intrins field coordin fre form algebra also discuss anoth gen typ holograph rel light front seem clos hooft origin idea holograph turn nat connect gen conceiv loc entrop quant form bekenstein class black hol surfac entrop
purpos short not annount result amount ver bootstrap liouvil the gen cas certain assum concern ex property fus transform assum charact fus braid coefficy solv system funct equ follow combin consist requir known result system equ un solv ir cent charg solv construct solv clebsch gord problem certain continu sery quant group repres construct assocy racah coefficy giv explicit express fus coefficy moreov express continu strong coupl reg provid solv bootstrap also reg
rel entrop mass fre boson field the study vary compact rieman surfac univers quant phys sign particul gravit phenomen exact express sphere obtain wel asymptot sery larg mass tayl sery smal mass also der exact express tor high gen howev asymptot behavio larg mass alway est const heat kernel method consist asymptot sery determin curv hent common homog surfac gen high expon van correct whos form determin concret topolog coefficy logarithm term sery giv conform anoma
show dond weyl cov hamilton field equ field writ duffin kem petiau matrix form consequ mod bet mat four spac tim dimend univers sign field admit hamilton form scal field view field theoret analog symplect matrix lead cal symplect structure also brief discuss could view cov poisson bracket giv bet mat correspond poisson bracket form hamilton field equ
crit examin rec suggest ambigu stat equ infinit quon stat describ dilut nonrel id extrem black hol show typ stat diff describ extrem black hol term ambigu stat apply
comput spectr sev crit amplitud dimend is model magnet field transf matrix method three lightest mass overlap spin energy op comput lat wid extract continu result also tak account correct scal irrelev op contrast prevy mont carlo sim fin result perfect agr predict matrix conform field the also obtain amplitud sublead correct matrix predict obtain
discuss poss gen calab landau ginzburg correspond gen class manifold spec consid ferm typ hypersurfac vary valu loop bet funct sigm model neg expect the mass howev quant cohomolog rel sigm const sigm suggest addit mass vacu ex remain massless sect the assum massless sect describ landau ginzburg the cent charg chir field charg comput topolog inv ellipt gener us the mass vacu comp geomet dat find result agr ev ev cas hypersurfac spin structure thu find evid geometry correspond cas spin manifold
correspond izergin korepin matrix three diagon solv mat bound yang baxt equ us mat construct transf mat op integr quant spin chain transf matrix correspond id matrix known symmetry argu transf mat correspond mat also symmetry nonstandard coproduc brief expl consequ symmetry
der vert abel stok theorem wilson loop cas mak coh stat repres coset spac flag spac consid yang mil the maxim abel gaug brok first show abel domin string tend follow theorem abel project effect gaug the der auth next independ combin abel stok theorem novel reform yang mil the rec propos auth process der are wilson loop four dimend yang mil the maxim abel gaug ow dimend reduc plan wilson loop least funda repres four dimend yang mil the estim diagon abel wilson loop defin dimend model der show funda quark confin singl specy magnet monopol origin are rel geomet phas wilczek holonom calc perform us instanton calcul dilut instanton approxim us larg expand lead ord
car study noncommut vert adhm construct instanton propos nekrasov schwarz noncommut describ algebr op act fock spac adhm construct instanton look zero mod dirac lik op feat peculi noncommut cas zero mod project stat fock spac mech project clar gaug group also construct zero mod gaug group demonst project also occ mech simil cas phys interpret project matrix model brief discuss
study modul spac superconform field the cent charg slight emend glob describ find loc vary known model compon assocy surfac among orbifold the obtain tor compon tri found play domin rol obtain field valu direct exceiv div ar orbifold prov dual orbifold der form pur within conform field the gepn model orbifold find loc prov isomorph nonlinear sigm model particul prov gepn model geomet interpret ferm quart target spac
expery research known topolog recurs rel gener zero comput point descend gromov wit inv arbit degr valu result consist virasoro conject also lead explicit comput hodg integr gener also der recurs rel simpl hurwitz numb simil grab pandharipand
superspac flat condit equ field equ supersymmet yang mil the dimend us der triv class solv rec mod flat condit propos explicit integr bas break symmetry artic investig phys cont group algebra method develop allow der phys field equ mot superfield expand supercurl system integr superspac constraint ident drast reduc field cont unconstrain superfield leav spectr includ origin yang mil vect field complet shel weak constraint giv ris addit field obey first ord diff equ geomet cov superspac constraint descend trunt wit origin linear system particul paramet famy light lik ray
construct deform quant algebr assocy group cas replac quant group bicov calcul deform easy allow includ cur algebr left right inv form examin poss gen gauss commut rel gaug the bas
pap giv complet selfcontain proof result annount show renorm quant field the spec inst gen mathem proc extract finit valu bas rieman hilbert problem shal first show quant field the combin feynm graph giv ris hopf algebr commut algebr dual hopf algebr envelop algebr algebr whos bas label partic irreduc feynm graph bracket graph comput insert graph vic vers correspond group group charact shal show us dimend regul bar unrenorm the giv ris loop qquad smal circ complex dimend around integ dimend spac tim main result renorm the evalu holomorph part birkhoff decomposit begin analys group show sem direct produc easy understood abel group high triv group clos tied group diffeomorph analys lat group wel interpret renorm group anom dimend cont second pap overal titl
easy chain fre self energy rainbow obtain contribut anom dimend also easy res rainbow fre self energy chain tam combin explod poss nest chain primit self energy diverg much demand problem solv term coproduc delt antipod grad op hopf algebr undec root tre vit op star star produc effect delt perform loop borel resum bphz subtract yukaw the spacetim dimend triv scal the encount residu star involv prim digit ev larg yukaw coupl precid resum remark loop calc suggest ord
dual integr system ar context seiberg wit the show funct indee seen gen funct mut poisson commut hamilton dual system demonst thet funct coefficy expand express entir term ordin seiberg wit integr system thu hamilton dual system
describ ext noncommut tor approxim tow finit dimend matrix geometry approxim carry ir rat deform paramet embed algebr noncommut tor approxim finit algebr construct rig der rec discret noncommut gaug the us finit dimend matrix model show prec continu limit model must tak clar vary aspect morit equ us form describ apply noncommut yang mil the
review rec work chir extend wznw phas spac describ extend bas field gen monodrom wel us bloch wav diagon monodrom symplect form extend phas spac invert cas chir wznw field found satisfy quadr poisson bracket rel charact monodrom depend exchang mat explicit express exchang mat term arbit monodrom depend form appear chir wznw symplect form giv exchang mat gen cas shown satisfy dynam gen class mod yang baxt equ poisson groupoid construct encod equ analog group encod class equ arbit simpl group exchang mat exhibit correspond poss structures admit symmetry
comput part finit temp effect act mass dirac ferm pres dirac monopol confirm induc charg temp depend effect act find infinit sery viol term gen famili zero temp tild term result analog rec result concern finit temp induc chern simon term
follow gen principl noncommut geometry poss defin met spac pur stat noncommut algebr gen coordin met gen us rieman investig gen property met finit commut cas correspond met finit also giv exampl comput commut noncommut cas
casimir energy dilut dielect cylind light veloc surround med evalu exact first ord num high ord first part carry us addit formula bessel funct deby expand requir
discuss shuffl id feynm graph us hopf algebr structure perturb quant field the concret exposit discuss vertex funct massless yukaw the
