これが work?あ、OKSo, I gave the definition of Kuromonach as a spectrum of its equivalent homology of certain space.So, now I give a classical description.So, approach is very different from the physicist's approach.So, physicist start with classical description and then try to quantize.But I just start with quantization and go to classical limit.So, the statement is like this.So, MC is birational to T cross T check,T check cross T divided by wild group.So, T is maximal compact subgroup, maximal torus, sorry.And T is its real algebra.And T check is dual torus.And Y double is wild group.So, this right hand side is usually called classical description of physics lecture.So, here is the proof, sketch of proof.So, we first know that equivalent homology for group G is just equivalent of homology of the torus for the same spaceand take the wild grouping variant part.There is natural action of wild group on the space induced from the group actionand they take the invariant part.So, this is some general result for equivalent homology for any G spaceand if you consider G equivalent homology,it is wild group invariant for equivalent homology for torus.Then you apply next, you next apply localization theoremequivalent homology.So, this is also very well known result in equivalent homology.So, it go back to the bottom,well in between and others.So, what does it say?So, you consider the torus equivalent homology group.So, this is a module over the equivalent homology of maybe T of point.But this is maybe I mentioned sometimes.So, you consider the same as equivalent homology just constant groups.And then you tensor with fractional field of this.And on the other hand you consider the equivalent homology of the fixed pointand similarly you consider.Why?But this is since torus act trivially.So, this is the equivalent same as ordinary homology.But anyway, you just then you have the inclusion homoisminduced from the inclusion of fixed point.Then this becomes isomorphism if you tensor this fractional field.So, then what is this?I mean you must also need to check that this multiplication for hereand multiplication for here and multiplication for here is compatible.But after checking also.But after checking this statement and taking the spectrum.So, it means that the spectrum of this is the while group quotient of spectrum of this,which is birational to spectrum of this.So, then you just need to understand what is the T fixed point.So, I think you will see if I describe the fixed point.So, R is a modular space.Flaved bundles, G bundles plus section.So, P and S and Phi.Such that Phi. Phi is the trivialization of P at over the puncture disk.So, you require one condition.If you send the section S by Phi, then this is regular.So, this is the condition.And you consider the T fixed point.If you are familiar with the modular space of torsion free ships on the surface,flaved ships, then you look at the similar fixed point.First of all, the T is coming from the chain changing of Flaving, Phi.Then T fixed point means that this P is T bundle.Or maybe if G is a general linear, we consider the vector bundle.And the T bundle means it is decomposing to direct sum of vector bundles.Aligned bundles, sorry.And S, S is section.So, section of.Now, the section has value in N.Instead of N, we take fixed point in T and T-valued section.And Phi is just, now it is trivialization of T fixed point.So, this is the description.So, it means this RT is nothing but R for.So, G is replaced by T and representation N is replaced by N of T.So, in this case, the value fixed point is nothing but the special case.Special case of our construction.So, group is replaced by torus and representation replaced by T fixed point.But this NT, because of the definition.So, NT is a trivial representation.So, you remember our computation.So, I give example.Example 1 and 2.Example 1, G is called T and N is called 0.And G is called T, not T, G is C star.You call C star and N is called C.And so, this, the first one is just the product of C and C star.And in the second level, we get C, C2.And non-trivial multiplication appears.So, you consider.So, in this case, you get prime R1 and prime R-1 is equal to W.And this appears because this condition plays some.So, maybe I do not draw this picture.So, you have here N equal to 0.And N equal to 1.And from here, you have some different fibers.And that was the reason why we get this W.But if this is trivial representation,so, this condition is automatic always.So, it means you can forget this trivial representation.Curon branch MC for G called torus and N equal LT.Maybe, I should not equal.So, I replace this.Then, this is just, I have T, T check and the algebra of T.So, this T is equivalent homology of point spectrum of this.And this is just pi, pi.Yeah, this is the same as the explanation.I find grass mania.This is come.So, ZN something.N is the rank of T.So, this is the proof.I think this is interesting.I mean, the proof is very simple.But somehow, it is interesting that this classical picture,classical distribution of the Curon branch appears as application of the fixed point.Action of wire group.I mean, this is a standard.You also need standard wire group action on this torus.So, diagonal action.Okay.So, then, I slightly define this, this, this, maybe,and really find the concept which appearsinputs in here.So, I have equivalent homology of point.So, this is h star.And this, there is algebra homoism.So, this is, if you have a function here,then you just send it to application to the unit.So, unit is, by the way, the five order,the unit element in the affine-rasse manning.Then, take a spectrum.So, you have mc.So, I denote this by pi.And I have spectrum.And this is, as I said, spectrum of h star t of pointright by yw.And this is T mod w.But it is also known that this is a reflection groupand this, this is just vector space.And this homomorphism, this survives in quantization.So, namely, h star of g cross c star of pointis send it to h star.And so, this is quantization.Cantized Coulomb branch.And this is commutative.This is algebra homomorphism.So, this is commutative.So, maybe I didn't mention.Yeah, I think I explained in the example.So, this is a non-commutative algebra.And this is commutative.So, this, in fact, this is injective.And this is commutative subalgebra.So, it means if you computePorson bracket for this kind of function.So, you just pull back these two functionsand take compute the Poisson bracket.But Poisson bracket came from the commutative.First, you lift the non-commutative.Algebra and take commutative.And divide by h bar and send h bar to 0.That was the definition.But here, F and G is just automatically lifted in hereand send it here.But that is commutative.As I said, this is commutative.So, this is just 0.So, it means any function factors through herePorson commuting.And also by this observation,which I just am erasing.So, I have mc and by pi.So, I have d divided by w,which is C of L, t cross t check,divided by yl group.And I said, this is birational.And this birational model is just givenby localization equivalent model.So, this is commutative diagram.This one is fast projection.So, in order to study the fiber,genic fiber,so, you just switch to here.So, this is digital torus.So, this means bar pi iscompletely integral system.Do you feel sense?So, you have sympractic.Yeah, maybe I didn't mention this.Sympractic variety.At least regular localization is sympractic.So, this is sympractic.And you have vector space to half dimensionandPorson commuting in the sense like this.Or maybe you just take coordinate entries,entities of these vectorsand you take commutative that punishes.Then another is generic fibers.Well, I mean, usually when peopleusually assume the generic fiber is compact torus,but this is algebraic one.Or maybe complex algebraic one.So, in this, this is integral system.And maybe just remark.Yeah, in fact,so, this construction works for K-3 version.And I asked several people whetherthis integral, if you notice this integral.But it seems it was unknown.So, maybe I'm a little bit confusedwhy it was not unknown.But, so anyway,so, from MC of K,so, this is spectrum of K-cell UGO.Then, so corresponding target spaceis spectrum of equivalent K-cell UGof G of point.Then, this is,so, similar to the equivalent homological point.This is also well known.So, this is just T divided bysimilarly this is.So, as I said, this MCK is something similarto Hitching.Hitching consider.But this is very differentfrom Hitching system.Hitching integrable system.Hitching integrable system existsonly for the specific complex structurewhich we missed.And so, it is defined.This source and source formand also the target spaceis vector space.So, the integrable systemis just given by the character.It's polynomial,coecent of character polynomialof the Higgs field.So, this is vector spaceand also the generic fiber.This is aberian variety.So, in this sense,of course, this integrable systemshould be much, much more interestingthan our interpretation.This seems to be much moreelementary integrable systembecause fiber,fiber is just as you told us and there iscomplex structure is fixed.But here, aberian varietychanges the complex structure.This seems to be much more interesting.But still, you see this existence.And it seems for character varietyyou still have this kind ofwhat do you mean by similar?Underright?Yeah, that is true.Yes, yes.In this case,Hilben of C cross C star.Then you just project to C.That is the integrable system.Symmetric polynomial.So, it is similar.And Hilbetsky,maybe I don't have timeto explainhow Hilbet scheme appears,but symmetric powers ofN of C cross C star will appearas a Coulomb branch,particular Coulomb branch.And then you see thatthis is exactlyin this particular example,this is the integrable system.In this case, symmetric powersof C star cross C star.Okay, thennext we will explainquiburgase theory.Quiburgase,so far,we just start withgeneral group and representationand definition is very general,but so I specializeto the case of the so-calledquiburgase theory.Then somehowwe have somebetter understandingof the Coulomb branchcompared with other cases.So, I willI'd like to mentionwhat we knowsome conjecture.Quiburgase.So, if you are familiarwith the definition ofquibur varieties,then quibur varietyis a Higgs branchof quiburgase theory.So, the first starting pointis very similarto the definition ofquibur varieties.So, q0, q1 is quibur.So, q0 is aset of vertexand q1 is a set ofarrows.I take two vector spacesvw,finitegeneral,complexvectorspaces,q0 graded.So,v is direct sumof vi and w is direct sumof wi.Then I takeg to be productof gl of viand then is aspace of framequibur representation.So, that is just linear spaceof h to h.So, h runs in arrows.So, you have vi,vj,wiand wj.Like this.And as I mentionedthat if you considerhyper,here,higgs branchwhich is n plusn stardivided by g.So, this is calledquibur variety.Sometimes you takecharacter of gand takegit quotientbut I will notenter that story.So, that is calledquibur variety.So,but ourinterest isin kooloom branch.So, kooloom,what is kooloom branch?The theorem,so this isdue to myself,myself isbraver when thinking aboutmyself.So,supposeq istypeadso,adgin kindfiniteadgin kind.Then,I have correspondingcomplexreductivegrouporsimple group.So,I takeadjoint typethe oppositeof thesimply connected one.Then,mcisisomorphicSo,I'm cheatinglittle bit.So,I will explainlittle bit morepossess statement afterwards.Modular spaceofgqmonopoler3withinguralityat 0.Type ofinguralityat 0isgiven by.I will not describethe detailor the distance.But,by thosedimension vectorsand alsothere's notionso-calledmonopole charge.So,this is determined bydimensionfororI.And thiskind of resultat leastintype Ayou candelivethis kind of resultin a physical wayso,using brainso,due tohandling andI'mnot surebutin general casebut anywayso,this kindof statement isexpected.So,monopole issome partial differentialequation whichisgiven bythe reductionof theinstant equationbyimposingtranslation invariant.Andso,thisin generalso,we allowsingulateat 0 alsothere'sso-calledbehavelikerbyrandsomethingand the type ofthispart isspecializedbythewIandformonopole chargesso,it isbetter toconsider thecasew equal to0.so,ifw equal to0,there'sso,this isduetomonopoles4R3so,this isduetohitchingand othersI thinkwhich arethis.so,this isisomorphicto thespace ofbased mapsfromp1toflagvarietyfullflagvarietyand thismonopole chargeis nothingnn isrankwhich is justnumber ofparticesofquiburnso,dimensionvican beregardedthat'scharge.and I saidthatI'mcheating a little bit.In fact,we provedisomorphictogenarizationof this description.and thisis knownonlyforthe casewithoutso,in that sense,I meanthis is notbesidesthing whichwe proved.anyway,so this issomethingtechnical.we thinkthis isgenarized.toso,this is alsoso,we considerjordan quibaso,this isyou haveonly one agentyou haveone-hitchconnectingthe vertex to itself.thenwe onlyconsidertwo-vectorspacesandwe considernartsand monophysmvvand homeofw and band gisglbthenin this caseand I assumethis isdimension kand this isdimension rso Higgs branchisthe modular spaceof rank rtotion freesheets onframetotion freesheets onp2with c2kbut Coulombbranch isdifferent.so mcisjust asdimensionsymmetricpowersofcasesymmetricpowersofxrwherexrissurfacexyisequalif thev hasdimension1dimensionvisequalthenthisisandsoas I mentioned alreadyso for trivialyou canignore trivial representationthen you justconsider this caseso thenit isthe same asexample 1 and 2and 1is a case ris equal to0and this isrpositiveand I mentionthat requal to 0case you getc cross c starand rpositive casec2 and simple typebasicso this isgeneralizationand 3so this is joint workis takayamaso q isaffine type nthenmc ismodular space ofso if youfrom dinkingif you replacefrom dinking diagramto affine dinking diagramyou go toone dimension upso instead of monopolesyou getinstantonmodular spaceofflamedso this casetype nso u nplusmultitablet spaceso this isso this isthe spaceas a compleximplicative varietythis issame asxrbut xrplussome knownflatsome knownflat metricon this spacehypercare metricso thenotion of instantondepends onthe choice ofremand metricin particularturbunate metricand euclidean metrichave differentsympathetic behaviorbecause of thismodular spaceis differentandthis ris in this casejustremandthe sumtotal sum ofremandand there is somediscrete dataso thishasdiscrete datadeterminedbydimensionsbasically thereis achanclasses andsomedatafor theflamingflamedunso this isunitaryunplusmultitabletso this ismaximalcompactsubgroupinglplusanditscompactsoII don'tI don't explainthehow theI meanI will justbriefly mentionhow proof goesand whywe havedifficultyto generalizethis resultfrom affine typeto other typessostrategic proofso we havemcandt-crosst-checkwandwsofirst stepfirst stepaskvisistwhat should becandidate of mcso we don'thave anygoodmethodsto knowwhat should becandidatesometimeswe couldnotnotour technologyis very muchlimitedso for exampleI meanI is a good friendthensoyou checkfor this candidateconstructso thissecondconstructthis integralresystemthennextandunderstudy fiberssofirst of allyou needto checkthatgenericfiberisthisArgevectorusso thenso you constructdefinevirationalisomorphismcompatiblewith thiscandidateandfour stepis the mostdifficult onesoshowcandidateisnormalandthiscandidatemaybe Ididn'tsay thatthis integralresystemhavesame dimensionso this isflatflat in thesense ofArgevecutumentso this ismost difficultdifficult anddelicateand fifthand ifso thiscandidateissomethingwhichhasbeenstudiedalreadythenthis kindofresultmightI meanif thisisreallyimportantspaceingeometricrepresentationstoriesthenthis kindofstatementhassomeisignificancetoimaginethis thaschamber structureandin generalbecause of thebecause wenot only the groupwe alsoadd the representationsochamber structuredepends ondependentononGGG andalso representationso thishipplaneis notnecessarilyhipplanein somecasehipplaneand thisfor herethe fiberso thisso fiberbut hereis torusthenyou show thatacrossthesuch kindpointso namelythe pointwhich liesin single hyperplanenot the intersection of hyperplaneacrosscodimensionto locusI don't knowwhat is right Englishor notso anywayI explainedso thisitacrossthisbut by theapplicationof thelocalizationcellso thisthisthis isbasicallyenoughtounderstandcool onbranchforrank onegroupssemi-simplerankrankrank oneplusrbtimesrbso likesowe know thatwhat ishowmcbehavesin thiskind of locusand also we know thatin thisexamplewe studywhat happensif youconsider thefiber over herethat's thekind of studythenthis impliesso thisis thepower ofalgebra presumablyso bycombiningthis normalitythensoforallthestatethecell andall theresultexjectin this mannerso somehowwe really needto knowthat what thecandidate andif we reallynotto show thatthis stepand sowe havealreadycandidatefor this casebut wedon't knowthe normalityandgeneralizebut weneed toslightlygeneralizethe definitionsothat isanother storyso Idon'texplainlet's seethat isalsoconsequence oftheI thinkbasicallyyeahso Istudiedthe equivalenthomology oftheandso Ihavealsoshow thatthis isfreeoverequivalenthomology ofpointso thisissomething likethis kindworkok inthe remainingminutesby thewayso oneinterestIhaveoneinterestingremarkso Isaidthatthis isinstantmodularspacestabnotandthis isxrplussomemetrichyperketometricwhich isnoteuclideanbutit isalsoknownthatthis isinstantmodularspaceonxrwhich isseatedbysomedominanceconditionis satisfiedthis isconditiondimensionvI-1plusdimensionso inparticularyoucomputethecharacterso wehavesisterand inthiscaseg2brplusoneynplusandthis isgiven bymonopoleformcharacteroftheinstantmodularspaceisgiven bymonopoleformbutit isalsoknownthatthis isneclosuppartitionfunctioncaseelectricnetpluspartitioncombinatoryform which isbutsomeovertuple ofyoungdiagramsbuttheseformular arecompletelydifferentso this issomewhat do yousayfinite sumfor ifyou fixthespaceand this isinfinite sumand this ispositive sumand this ispositivenegative sumbymiraculousreasonso thesetwoformular aresameso Idon't knowanycombinatoryproofofthis statementbutsomehowthroughidentificationof thecoulomblunchIobserve thisI'mnot surehow tounderstandwhat happenswhenOKthen remainingfive minutesI will mentionsimpleactualityso as I saidwhen thesimpleactuality is rapidlygrowing soI don'tI cannotsay muchbutsomehowI hopeI hopeone resultwhichgives yousome flavorofsimpleactualityso there isりかたプラウドフットウェブスターSo they are basicallyintested in therepresentation theoryof quantizationso butthere aresome othermore geometricstatementso foranother exampleisdue to Hikitaso maybeand there aresome othersso if yousearchin thekeywordsimpleactualitythen maybeyou findso oneI assumethe followingso supportsmc andmhhavesimpleacticresolutionso Idid notmchso I didn'tI didn't mentionat allforso for mhit issomehow well knownthat if youconsidertheGIT quotient insteadof netsimpleacticresolutionand formcI don'tmatch at allbut there aresomestandardconstructionsome constructionsimilarmodificationandthismodificationmakes sensebutsometimesit hasonly givespartialresolutionnot fullsimpleacticresolutionbutsimpleacticand this istrueforquiburgaseofrfinetypebut if you go tothe typed ordefinite casethen this isnot truein generalso this issome assumptionthensoconjectureinsimpleacticdarity saystheoiler numberoftheconjectureso let uschecksimplestcaseand forrfinetypeI can computeboth sidesand they matchso this is trueso I justcomputefor thesimplestcasefor thecase ofjordanquiburge isoiler numberwhat isyour notationfor e?e same?canconsiderthis issimilartothefirst testforthemiracymmetryusualmiracymmetrysoyoucomparetheoilernumberforxand itsmirarsotheyaremineequalup tosigniswhatnomemkrsothis ismodelisofflamedmotionfreesheetsonp2equaland c2itsgotkand mcchilderso this isjustimentionedso this isresolution ofc2divideso by the assumptionso I excludethe definition ofhiggsbranch isthrightlyproblemin this casemaybe youseed this caseI assumer is positiveand this isresolutionand there'sstandardonehealk ofso you firstresolvethesimplesingularityand takehealvert screenso this istheresolution ofsimilarp2 Equalp2p1p2p1p1p2p1p1p1p1p1p1p2p1p1p1p1p1p1p1p1p1p1p1p1p1p1このように私が言ったメクロスパティションファンクションは、この2つのプロビアムダイアブラウンスが同じです。そのため、この2つのプロビアムダイアブラウンスが同じです。ご視聴ありがとうございました。