Gの意識を表して、この方向を見ると、モジュライスタックを表示します。モジュライスタックのGバンドルとNバリードセクションを表します。D、D、D、Dです。スペコオ、D、Kのスペコオ、そして、G-Bundleのディスクは、Trivialのバンドです。このバンドは、Trivialのバンドにアイソモフィックです。TrivialのバンドはTrivialのバンドにアイソモフィックです。そして、このバンドは、P1、S1、P2、S2のバンドにアップディスクを使います。P1は、G-Bundleのバンドにアップディスクを使います。P3は、G-Bundleのバンドにアップディスクを使います。そして、I have phi, which is identification of P1、P2 over HOMAL-DISK, such that phi of S1 is equal to S2.そして、I have phi, which is identification of P2 over HOMAL-DISK, such that phi of S1 is equal to S2.この技能、S2は、Punctureのディスクに、パンクチャルディスクを使用しています。そして、P1は、P2のディスクを使用している可能性があります。ご視聴頂き、このディスクは、P1のディスクを使用している可能性があります。しかし、一部の定規は、この部分を定規すると、この部品について定規することができます。このように、この部品について定規することができます。このRは、最初に定規することができます。この部品は、G-Bundleのモデルアイスペースで定規することができます。この部品の定規への定規を定規することができます。2Lは、G-Bundleの定規の定規の定規を定規することができます。S2のファイアウェスはS2と同じです。S2は本来について定義すると、Nの定義の部分について定義すると、この定義の定義が定義されています。この定義の定義はRです。ではこの定義の dressing is also using the own things.今日のこの定義の定義は、フィックスの定義の部分を定義しましょう。的話私は定義をすべきに忘れ thin内だと思います。 サブマイモクの仕事を瀬優に集ます。これはのじグループ。ボレルム・ホモロジュブと言ってこれはプロダクト・マルティブリケーションを購読しましたコンボリューションプロダクトコミュタティーですクロムランチはアファイン・アルジブックスキームを作っていますしかしこれはファイネテリーゼネレーターですカントジーションをマルティーゼネレーターですキャント死 Huレウソンより厄介ですアファイネテリーゼネレーターですなぜあれのコンボリューションはアファイネテリーゼネレーターですアファイネテリーゼネレーターはこのアファイネテリーゼネレーターが結構半分你在地なのでアファイネテリーゼネレーターが詳細して presentセミダイレクトプロダクトに関しては セミダイレクトプロダクトに関することができます今 クロンブランシーの種類を説明しますフィルックスレッドレッジの種類は 多くの種類が説明されているのですがこの説明はこの説明の説明の説明を確認します次に  Major LPの種類を 確認しますI don't give the precise definition of the homology groupThis is a slightly cheated notationbecause this grading is not negatively graded because this is the infinite measureI must be a little bit careful about definition of the gradingand this is both positive and negatively graded私を信じてください。これはG-Gradianです。このMCOIHyはGradianアルジェブラです。このアルジェブラはコンパティブルです。このMCOIHyはC-Stockです。このアルジェブラはコーディネートリングのMCOIHyです。このGradianアルジェブラはコーディネートリングのコーディネートリングのコーディネートリングです。後、この開開始するクリスマネートの違いしてナウンホロモーフィックの意見ですが、S原型製裁多めでそれを確認させます。しかし、このように、このコミュタティブリーマンを取り組んでいます。特に、ヘイパーケラメトリックにはアクセスがありません。このSTU2のアクセスは、コンプレックストラクションはありません。S1のアクセスは、S1のコンプレックストラクションのアクセスがありません。ディフ advisRってありません。因績がどうしても全部ald nが multサイドを解決できません。単書ではですね。その限界はではないのです。ここではでも、욱でみかんの判定の方を確認しています。その方を受け出すよです。ここでは、 пойは尺の面を伸ばしている方を受け出すと、エクセスは、コンプレックストラクションの・・・スゼムを拒めると、邪魔なさはアクセスしてます。ここでは、すべてから尺 sls projectent で、これが最初の特徴です他の特徴はアファイングラスマニューのファクトを使用していますGOOボウはアファイングラスマニューのファクトを使用していますこれがデータを使用していますオビッドはアファイングラスマニューのGOOボウでディスクリートを使用していますこれはドミナントコウエイトで使用していますコヴェイトは just a group of homomorphism from C star to T.T is maximal torus.So, for example, if G is a general linear group, T is just a diagonal matrix.And homomorphism T is...So, I usually do not call it lambda.So, for G, G is GLN.So, this is lambda of T.Just diagonal matrix T.And you have a wild group action on this.And the dominance condition means that you choose a particular representative in the wild group orbit.And in this particular example,so this condition means lambda1 and lambda2 is in decreasing order.So, this is the dominance.For such, so lambdai is integers.For such a data, you have...So, you can replace T by G.So, you can consider this is a matrix-valued function over the puncture disk.And you can regard it as a point in the affine glass mania.And orbit is just orbit passing through this point.So, it is well known.And so, if you are familiar with the glass mania and orbitunder the parabolic subgroup,then this orbit should be regarded as an analog of Schubert cell.So, you have the cell decomposition of the affine glass maniaand you have similar decomposition in the usual glass mania.Schubert cell is decomposition of the usual glass maniaand affine glass mania has similar decomposition.But of course, this is the infinite dimension.You don't have finitely many cells.Instead, you have infinitely many cells.And it is known that the Schubert cell is called...This is the cell and this orbit is not really cell,but it's very close to cell.So, I belong to this orbit in this way.So, you have projection to a partial flag varietycorrespond to lambda.So, P lambda is a parabolic subgroup correspond to lambda.And if, so for example, this, I guess,to this kind of...So, according to the...if lambda says several entries are equal,then you write the block diagonal matrix.And if all the entries,all lambda's are distinct than just upper triangular matrix.But in general, if you have multiplicities in thispartial flag variety.And this is vector bundle.And our space R,because R is a modular space of pairs,bundles and sections.So, you just forget the section.So, we have projection to affine glass mania.You restrict your G lambda,then I denote inverse image by R lambda.So, this is again vector bundle.So, in particular,so as the space R is the compositioninto the dominant co-weight.So, R lambda.So, this is total space of vector bundle.In fact, this is infinite rank vector bundle.So, you must be a little bit careful about it.But it's not so difficult to justify the argument.Vector bundle over G lambda.Then, this implies...So, this homogenous,this partial flag varietyis well known to have only even degree homology group.And you consider this as a stratificationand compute homology group of this oneby my vitals.Then, you see this one calipolynomialof this homology group.It's written in the following way.So, sum over lambda.This is the dominant co-weight.And you consider one calipolynomialof H star of G ofG monopy lambda times T to the delta of lambda.So, in the usual finite dimension case,delta lambda is basicallythis rank of this vector bundle.And, as I said,this is infinite dimension also.You must be a little bit careful.So, that degree is defined so thatthis infinite dimension vector,you compute a rank.There is some...So, R lambda is a sub bundleof some ambient vector bundlewhich is,in some sense,vector bundle over the affine glass mania.And this R lambda becomes vector bundlerestricted into this close,locally close sub-variety.And the difference of the rankfor some reference vector bundleand this vector bundle is finite.And this delta lambda is basically that number.So, anyway, this is well defined.So, in particular,this implies thatthis one calipolynomialcan be computed.This has...So, this is well known.So, this is isomorphic to H star of P lambdaequivalent homology point of this.And this is well known.And from this definitionof our Coulomb branch,this is left-hand sideis a character of this c starwith risk to c staraction given by homological degreeof this coordinate ring.As I said,some combinatorics.And this formula...So, it means you have the formulafor the Hilbert series of Coulomb branchgiven by this kind of combinatorics expression,some over the dominant overweight.So, this is called monopole formulaand by Coulomb and Zaffaroni.So, in fact,I was partly motivatedby their work.And they have some formulafor this Hilbert seriesof Coulomb branch.So, I thought thatthat should be the homology groupof some varietyand I tried to constructvariety so thatis one calipolynomial reproducedthis monopole formula.So, this is another evidencethat our definition should becorrect definition.So, I continue other properties.So, this is called fugacityfugacity in physicsrelation.So, you considerconnected component of Rand thisisprojectedaffine glass mania and fibrousif we restrict fibrous vector spaceit is connected.So, this is same asconnected component of affine glass mania.And as I saidthis is pi zero ofmap ofholynomial mapof s1 to gcby by gcor based maps.So, and thisinterpology, this is well knownthat this isisomorphic tofundamental group of gand the multiplication.So, it means thath starg oforSo, according to the connectedcomponent of this decomposehomology group decompose, this ispi1g gradedand this is againcompatible multiplication.So, this is graded algebra.So, again you canconsider the actionof the groupon thespectrum.So, in the previous caseit meansthat the c star actionand in this caseso, you havethe action of the so calledponterecine dual.So, this is justhomology ofpi1g to c star.So, if g issemi-simple,pi1g isfinite cyclic group.So, it's pankare dualponterecine dual isfinite groups.So, in that case it's not so interesting.But, for example, if g isg is c starthen pi1 of g isz and it'sponterecine dualis returned back to c star.In fact,well, naturally thisshould be regarded asdualtourus of c.So, in this caseyou have the action of torus.So, it's an interesting case.So, weI explainedyesterday the example thatwhen g is c starthen equal 0and then equal c.So, in this case thecouron branch is c crossc star.In this case, this is c cross c.And if you remembermy computation.So, I firstso, in this case Ibasically say thataffine glass mania for gis just gintegious.So, Iwrite this picture.So, this is g to the power n.And you can identify thiswith both.That is becauseI need somenormality.So, in order to extend itto ac interaction, I needsome renormalization of that.So, thisas I said, delta is basicallythe difference betweenthe vector bundle, R-Lambda andthe ambient vector bundle.So, this isnot negative, but becauseof this normalization you alwaysget.You get both positiveand negative.So, anywayI denotew.It's h starequivalent to homology ofpoint.And I denotefor this first case andR prime of n.And this hasgrade n.So, it meansso, for thisI denote x, x and y.x is R of 1 primeof 1.This is y isprime of minus 1.So, in this casethisw iscorrespondedfirst factor and second factorcorresponded to this.So, it meansthe c star action is justaction on the second factorin this first example.And in the second examplethis x is multiplied by1 and y is multiplied byminus 1.So, this ishypoblic c star actionon c2.Let's see.Okay, then next slide.So, I started withexplainingthat this isthree-dimensional n equalfour supersymmetric gauge theoryassociated with thegroup and itsimpective representation.But I also mentioned that there'sfour-dimensional theory.So, this is the kind of remark.So, recall g and m,in fact, also definesfour-dimensionaln equal to theory.And you have the notion ofCuron Branch IVCuron Branch IVCuron Branch IVR3Cross S1.So, if you know thatyou will hear aboutU planesU planes are some vector space and U planesiscalledCuron Branchbut it isCuron Branch IVinstead of R3Cross S1.So, on the U plane you havethis elliptic vibrationwhich will be calledin this language calledCuron Branch IVR3Cross S1.So, anyway, soyou have this one andwe have also the constructionCuron Branch.So, replaceequivalent homology groupof Rbyequivalent case theoryof the space R.Then still, so thisstill hasconvolution product.So, which is commutative.So, you can consider it's spectral.So, we denotemc of k.So, K-theoretic Curon Branch.So, this K-theoretic Curon Branchis in fact thethis Curon Branch.But you must be a little bit carefulfor this statement.So, this 4D Curon Branchexpectedhyperkeler.And in 3DSo, we haveSU-2 action rotatingallhyperkeler structure.But in 4Dthat is not expected.It's only S1.So, in hyperkeler money 4Dyou have complex structureparameterized by S2.So, like this.And S1 action rotatingjust rotating this sphere.And this isthis is similartosomething similar tomodular space ofhitching of Hicks bundlesover Lima surfaceinterested by Hitching.So, Hitching's modular spaceishyperkeler.So, this appears ofvector bundles andHicks field.And the Sist action is justnumber 5.But this is aparticular complex structurein S2hyperkeler structure.So, this Hicks modular spacecomplex structure as a Hicks bundleis correspond to eitherNorth Pole or South Pole.And this isalso in differentcomplexure.This is equalin different complex structure.Or, in fact,all other complex structures are equivalent.So, for any choiceof other complex structures exceptthis North Pole or South Pole.This isisomorphic tothe character variety.So, this isrepresentation ofPyram-Sigma to the groupconjugation.And inparticular, this is affine variety.So, in fact,the 4Dcelli which is obtained from6th-dimensional 2,0 cellby compacting the RiemannService, this sigma.Then you getthe 4D-dimensional cell which is not gauge-celli asparably explained yesterday.So, this, if you considerthe corresponding Coulomb branch,then this is the Hitch modular space.So, in that sense, this is not justsimilarity.In particular examplewhich unfortunately is not gauge-celli isgives this Hitch modular space as Coulomb branch.So, we expect this is truein general.So, inour definition, this definition,so bythis description,this is justaffine algebraic variety.So, we cannotas Iemphasize several times already.So, our definitioncannot be,cannot sayanything about hyperkeler structure.So, we cannot saythis modular space.But, somehowwe expectthe following.So, thisexpectation.So, we learnedthis expression from Gaiotto.So, thisK-celli to Coulomb branchisCoulomb branchin equal tosuper-symmetric gauge-celliwithgeneric complex structure.In some sense, thisgeneric complex structure is mostuninterestinguninteresting complex structure.So, if you know that thisreaction between the necrosispartition function in instanton counting.So, instantoninformation of instantonpartition function is encodedin thethis complex structure, this special complex structurewhich you can define, hitting vibration.And if yougo to a differentcomplex structure, thenthis complexvariety doesn't know anything aboutinstantonpartition function.So, this ismostunuseful.So, in order to say somethinguseful,I really need to understandthe other complex structureon thisand I don't have any ideahow to defineother complex structure at this moment.And in three-dimensionalcase, as I said, all complexstructures are equivalent under theICO2 action.So, I can pick upparticular complex structures.That is not sobutit's still.I cannot see thehigh-package structure, but I canat least see all complex structures.In that sense, it is not so, but infour-dimensional, it's not so good.So, anyways, I giveexamples, correspond tothe two exampleswhich I explained.So, oneis g called c star.It'szero.The space is verymuch similar.So,the space itself is the same.Aphangra's mania of g is justz.And I considerequivalentcase-ary ofg of o ofg.In this case,and this decomposesintoequivalentcase-ary ofpoint,multiplydebyr of n.So,rn in the previous examplewas in the fundamental class, butthis is just thestructures of the pointgn.Then,multiplycation for rnis the same.rn times rm isequivalent to rn plus n.Butthis part is slightly modified.So, this iswell known that this isthe character representation ring of c star.So, this isRolan polynomial ring.Maybe everything iscomplex.Complex, right?So, this is c.So, in the previous example,this is polynomial ring in one variablew.But now,w must be inverted.So, this isw is a standard representation c star.Wait one representation,w inverse iswait minus.So,this from thismcok is justinstead of c cross c star,you just getc cross times c cross.So, thisin this case,this is r2 crosss1 cross s1 and you haveflat metric.Andthehitching fibration in this case is justprojection to r2.And 5 by the tolerance.So, you mustsomehow identifyp-p-p period ofq and px.q and you consider this aselectricity.So, in the second example,so, g equals c star.n equals1dm.So,the space is something like this.So, over here,n equals0,you haveg.Forpositive,you have c ofg.But fornegative one,you haveg.Maybe in this example,theambient vector bundle is justc of this.I didn't meanwhat isambient vector bundle.So, all these fibers of ris alwayssub-bundle in theambient vector.So, k ofg ofris againn of gand k of g ofpoint prime of rn.So, this prime meansinstead of point,you justconsider the fiber.This structure is equal.This is again c ofdoubleinverse.Then,you needto compute the product of thisthis element.And as before,it is importantto understand the product of r1and r-1.So, in the homologicalcase,this is just equal todouble.That is basicallyyou just compute the Poincaredual of this sub-band.In the case,youmust replacePoincaredualby this kind ofstructure c and inparticular cossule resolution.So, instead ofw,so you get1-w.So,this comesfrom.Or maybeyou just cut cossule resolutionthe origin in c.Structure c for theorigin in c.So,you get this 1- So,Ijust let this asx and this y.So,this isis equal to cof xand yand wand winverse.The relation is xas I said,this is x,y.So,this is the descriptioncatery-coulomb branch.So,inparticular,this isspectrum.Ifx-spectrumr isxy in c2.So,you caneliminate variable doublefrom this formula.Butyou also need to assumethat1-xyis invertible.So,this is not 0.So,this is the condition.So,thenI can checkthe conjecture in this example.So,you look forhypercalametric on this space.So,this is verysmall just four-dimensionalhypercalametric.There are not so manyexamples of hypercalamane.In fact,it is knownthere is so-calledOrgulibuff metonic.So,this is a hypercalamanehypercalametric.This is a hypercalamatic on this part.In fact,so,OrgulibuffinterestedOrgulibuff metric.Again,interesting part isthe other complex structure.You can see theelliptic fibrations.Andin this case,Orgulibuffmetallic is alocal model for thisheaching fibrations,heachingmodular space.Andyou have singular fiber at theorigin.So,it isSo,insteadvector space,in fact,Orgulibufftheanalog ofheaching fibrations,target spaceis a disk.Andyou have theelliptic fibrations.And you havesingular fiber at theorigin.So,thisthisaffinevariety isOrgulibuff metricwithgenetichypercalastructure.Ishould look at thereference.So,thatis thereI think that is a zerosingulate?Yeah,I don't know.So,youjust look at.Maybe Itry.Yeah,I will explain tomorrow.I will recall tomorrow.IfI look at the paper.Okay.So,thisis aboutso,in the remaining ten minutes,Iso,I will explain therefinitionofKurom,hex branchfor the theorywhich has no Lagrangianso,the theorywhich comes fromSixth-manger type20 theory.So,this iskind of side remark that Ijust want to mention thisconstruction becauseit,in fact,ourconstruction is not reallytherequire the Lagrangian descriptionof this quantum field theory.So,thisex branch ofclass S theory.So,thisisfor the n equal to supersymmetric quantum field theory.Optained reductionor compactificationofSixth-2,0 theory.TOGbyElliman surface,possiblywith punctures.So,as I said,theKurom branchof this theory,for themanger theory,it's isomorphicto the Hitchin modular space.So,in fact,original theory,Sixth-manger theory is parameterizedby the real algebra,butonce you compactify,somehowyou need to choose the correspondingcomplexreductive group orcompactlygroupspace.No,no,no.I will talk about the Hitchin branch.So,that is some differentconstruction.So,in that sense,Idon't need to construct theKurom branch,but insteadI'm interested in the Hitchin branch.So,this is,in fact,askedby Mu and Tachikawa.So,they askgive amathematical rigorousconstruction ofHitchin branch,andcheck expected property.I will not explain what is expectedproperty.Basically,itexplains what happens when youpinch the lemur surfaceinto two pieces.So,thisHitchin branch isexpected to beindependent ofcomplexstructureofSigma.So,inparticular,becauseof thisexpected property,itis enough to constructSigma isp1 with0,0and 1,2.0 is1,maybe0 and3 functions.So,after that,you canglue lemur surface andcorresponding operationexists in this Hitchin branch.You can justconcentrate it.So,keypointis the following observation.So,forGage theory,sowe usethe space r,whichis the same.So,I denote this projectionby pi.Then,I considerequivalent homologygroup of this space r.Maybe,I also denote thisprojection to pointby p.Then,inorder to compute homologygroup,so,somehowin the last two centuries,peoplereally noticed thatyou're not just not onlyconcentrating this abstract homology.It is also better toconcentrate the relative homology.So,you can interpret thisequivalent.So,everything is inequivalent.So,youconcentrate the so-calleddealizing complex on the space r.And then,you push forward intothis.So,thisis a complexobject in thedelived category of thisequivalentconstructible C4R.So,if you are not familiarwith this language,just considerthis is a kind of vector bundlegeneralizationof local system.So,it is alocally constant vector bundle.So,it is a locally constantsubset.So,it is alocally constant subset.So,this is what Iprocessed.Then,if youhave a C4R,thenyou can push forward intoaffine glass mine and furtherto a point.Then,so,thisis some complex.If you take the homologygroup,then you recover thehomologygroup,theoriginal space.Then,youcanregard this astwo-steppopulation.So,p starof this pi staromega r.So,thisobjectis in thedelived category ofG of O ofconstructible of theaffine glass mine.So,ifyou just push forward toaffine glass mine and then you getover theaffine glass mine.So,ifyou decompose into orbit,theneach orbit you havelocally constant vector bundle.And there are someglued in some way.So,this is the objectin this category.So,and thiscategory hasbeen studied ingeometric representation theory.So,thisD ofconstructing.So,inthe usualdelived category,so you havesomeAberian category,andifyou have standardconstruction of derived category.Unfortunately,thisis not thederived categoryofAberian category.But somehow,it still containsAberian category.That iscalledPowerSheep,Category ofAberian Equivalent PowerSheep.Andthis is theAberian.Andmoreover,convolutionproduct givesmonoidal structure.Theconvolution product which Iuse to define the product.Andso,I said that this hasconvolution product.Andfor if Iexplain in this language.So,this piomegastar,pistar ofrealizingsheep is a ringobject,commutative ringobject.So,this is commutative.Thismonoidal structure is commutative.Commutative ring objectin thisdelived category.So,youhaveif I denote this byA,so you haveA,convolution productwith A itself toyou have homomorphism likethis.Andyou have,it is commutative and associative and so on.Then,in fact,you observe insteadof introducing space R.This is the key.MaybeI should stop soon.Itis enough togive commutativering objectin this categoryto defineA,or maybe in this context,this is Higgs branch.So,this is a new definition.New way of looking atthe Higgs branch.And as I said,this category,this category has been studied in geometric representation theory.And inparticular,so it isfamous fact.So,this is calledgeometric satake.So,thismonoidal category isisomorphic to the category offinite mental representation,Langrange DR group.So,this is the statement.This is very non-trivial statement.So,G,Langrange DR groupis definedfrom G in terms ofsome combinatorial gadget.But,somehow,thisgives moreconceptual explanationfor theconceptual meaning ofLangrange DR group.So,this category isAberian category.MonoidalAberian category,this isisomorphic to the category.And inthis category,so you can justso,inrepresentationof GL,or more preciselyits completioncontains a ringobject.So,themonoidal structure isjust given by the tensorring here.And you justconsider somerepresentation ofLangrange DR group,whichhas this homomorphism.So,this is justgivenby theregular representation.Then,the multiplicationof the group,Langrange DR groupgives this homomorphism.Well,this spaceis more difficult toanalyze than theLangrange DR group.So,unfortunately,wedon't know that.This isNormality we don't know.Even finitely generatednesswe don't know.What I payin this project.So,anyway,soI must finish.I need to conclude.Based on this regular representation,I canconclude.I can takebasically,some productin this category,I canconstruct newringobject,commitative ringobjectin this object,and then justapply this configuration.Then,it gives thehiggs branch of classes.So,I stop now.Thank you fortell me.