Quantum decorated character stacks
QQUANTUM DECORATED CHARACTER STACKS
DAVID JORDAN, IAN LE, GUS SCHRADER, AND ALEXANDER SHAPIRO
Abstract.
We initiate the study of decorated character stacks and their quantizations usingthe framework of stratified factorization homology. We thereby extend the construction byFock and Goncharov of (quantum) decorated character varieties to encompass also the stackypoints, in a way that is both compatible with cutting and gluing and equivariant with respectto canonical actions of the modular group of the surface. In the cases G “ SL , PGL weconstruct a system of categorical charts and flips on the quantum decorated character stackswhich generalize the well–known cluster structures on the Fock–Goncharov moduli spaces. Contents
1. Introduction 12. Decorated surfaces and stratified factorization homology 173. Monadic reconstruction of module categories 244. Charts and flips on Z p S q via excision 405. Examples 56References 581. Introduction
In this paper we introduce decorated character stacks and their quantizations using theframework of stratified factorization homology , as recently developed by Ayala, Francis,and Tanaka [AFT17]. Given a reductive group G with Borel subgroup B and its Cartanquotient T , a decorated surface is a bipartite surface with open regions colored G and T andcodimension one defects between them labelled B . The decorated character stack Ch p S q isthe moduli stack of G - and T -local systems over each corresponding region of S , togetherwith additional B -reduction data along the defect lines. The quantum decorated characterstack is a quantization Z p S q of the category QC p Ch G p S qq , obtained by integration over S with coefficients in a parabolic induction algebra , which prescribes the ribbon braided tensorcategories Rep q p G q and Rep q p T q in their respective regions, and their central tensor categoryRep q p B q along line defects.The resulting quantum decorated character stacks form a basic ingredient in the study ofBetti quantum geometric Langlands, a mathematical model for the Kapustin-Witten twistof N “ d Yang-Mills theory compactified on R (that is, evaluated on decoratedsurfaces times R ). In mathematical terms, the structure of the parabolic induction algebra issimply that of a 1-morphism between Rep q p G q and Rep q p T q , these categories being regardedas objects in the 4-category of braided tensor categories [BJS18] which is the target of the4d TFT, so that Rep q p B q defines a natural transformation between the G and T -theories.In physical terms, the parabolic induction algebra defines a domain wall between the G -and T -theories, providing the structure of local operators at interfaces of bipartite manifolds a r X i v : . [ m a t h . QA ] F e b DAVID JORDAN, IAN LE, GUS SCHRADER, AND ALEXANDER SHAPIRO colored by G and T , and hence our constructions give computations of global operators forsuch manifolds.In this paper we develop a number of tools which allow for very concrete computationswith quantum decorated character stacks. Our lodestar is the following proposal of DavidBen-Zvi: The quantum cluster algebras associated to a marked surface by Fock, Gon-charov, and Shen describe charts on a proper open locus of Z p S q , consistingwholly of non-stacky points. Let us briefly recall the quantum cluster algebra constructions of Fock, Goncharov and Shen.In [FG06], Fock and Goncharov consider a pair of moduli spaces. The first one, X G,S , is themoduli space of framed G -local systems on a decorated surface S , defined for an arbitrary splitreductive group G . The second one, A G,S , is the moduli space of decorated twisted G -localsystems on S , defined for any simply-connected reductive group G . Among other results, itwas shown in [FG06] that X P GL n ,S and A SL n ,S are respectively cluster Poisson and cluster K -varieties , and moreover, form a cluster ensemble. In [GS19], [Le19], [Ip18], these resultswere extended to arbitrary Dynkin types. In [GS19] the moduli space X G,S was promoted toa new one, P G,S , parameterizing framed G -local systems with pinnings. This moduli spacealso has a cluster Poisson structure, but in contrast with X G,S , allows for frozen variables atthe boundary of S and thus admits gluing maps that are not available for the spaces X G,S .Non-commutative deformations of cluster Poisson varieties were constructed in [FG09a;FG09b], by associating to each cluster Poisson chart a quantum torus algebra, and promotingthe transition maps between charts (called “cluster Poisson transformations”) to algebraisomorphisms between the skew fields of fractions of the corresponding quantum tori. Theoutput of this construction is best understood as a functor X qG,S from the cluster modulargroupoid , whose objects are cluster Poisson charts on X G,S and morphisms cluster Poissontransformations, to the category of quantum torus algebras with birational isomorphismsbetween them. One may then pass to “global sections” to obtain an algebra L q p X G,S q consisting of all quantum torus elements which remain regular under any sequence of quantumcluster transformations.Ben-Zvi’s proposal predicts on the one hand that the category Z p S q should capture therich combinatorics of the q –deformed moduli spaces P qG,S , X qG,S , and on the other hand that Z p S q provides a “stacky” enhancement of each one. Such an enhancement is important:whereas removing the stacky points allows for more explicit descriptions of open loci in clusterterms, it crucially destroys the topological functoriality present in our construction. Indeed,the assignment S ÞÑ Z p S q is a priori an invariant of surfaces, naturally equivariant for theappropriate mapping class group. In the traditional approach, the corresponding equivariancestatement can only be deduced as a consequence of the highly non-trivial construction of acluster structure on P qG,S , X qG,S .The second important reason to work within the framework of stacks is that doing so allowsto construct the moduli spaces and their quantizations in a way that is much more localwith respect to the decorated surface S . The central idea of the standard cluster approach isalso to work locally in S by regarding it as being glued from decorated triangles – these arethe smallest decorated surfaces for which the moduli stack has an open rational subvariety.Yet our techniques allow us to localize all the way down to the very simplest nontrivialdecorated surface – a disk D with a single domain wall between G and T regions. Then, in a Cluster Poisson and a cluster K -varieties are also known as cluster X – and cluster A –varieties, respectively. UANTUM DECORATED CHARACTER STACKS 3
Figure 1.
Triangle D with a gate in each T -region, composed of three copiesof D B via excision along G -regions. In what follows, we abbreviate overlappingregions by thick dashed and dotted lines, as shown on the right.precise sense (explained in Proposition 2.8), the parabolic induction algebra Z p D q “ Rep q p B q completely determines Z p S q for any other decorated surface S . In particular, the decoratedtriangles fundamental to the standard cluster formalism simply arise by gluing together threecopies of D , as indicated in Fig. 1.Finally – and most importantly – quantum decorated character stacks fit by constructioninto the framework of fully extended 4-dimensional topological field theory with defects, sothat in particular one can define functors Z p S q Ñ Z p S q from decorated cobordisms between S and S . These functors do not preserve – and hence cannot be defined via – the non-stackylocus. In future work we intend to explore applications of the functoriality properties ofquantum decorated character stacks to the construction of invariants of links in 3–manifolds,and in particular to the proposals in [Dim13; DGG16].Our main results in this paper establish Ben–Zvi’s proposal in rank one – i.e. for the groupsSL and PGL . We exhibit a collection of categorical open “cluster charts” Z p ∆ q Ă Z p S q , withtransition functors given by simple “cluster mutations” Z p ∆ q Ñ Z p ∆ q relating triangulations.In Theorem 1.19 and Theorem 1.21, we describe each chart Z p ∆ q explicitly as a category ofmodules for a quantum torus arising via monadic reconstruction, and we also describe flips ascertain explicit birational transformations between these tori. We show that in special cases(see Theorem 1.21, Remark 1.20, Remark 1.22, Corollary 1.25) our charts and flips coincidewith those on A q “ , S , P q PGL , S , X q PGL , S . We note that in the remaining cases – A q ‰ G, S , A q “ , S , P q SL , S , X q SL , S – analogues of cluster structures were not previously known, so there is nothingto which we should compare our construction. In a similar vein, we extend to arbitrary q the cluster ensemble map, which was previously only defined for q “
1, or in the absence ofpunctures (see Remark 1.23).For the remainder of the introduction, we give a more detailed overview of our basicdefinitions and main results.1.1.
Decorated local systems and their moduli spaces.
Fix a connected reductive group G , a Borel subgroup B ã Ñ G and its universal Cartan quotient T “ B {r B, B s .By a walled surface we will mean an oriented surface S , together with a collection C “ t C , . . . , C r u of non-intersecting simple curves embedded into S , such that B C i Ă B S and p C i (cid:114) B C i q X B S “ H for every C i P C . Walls and regions of the walled surface S arerespectively the curves C i and the connected components of S (cid:114) C . Definition 1.1. A decorated surface S is a walled surface S together with a labeling ofeach region from the alphabet t G, T u , such that any two regions sharing a wall have distinct DAVID JORDAN, IAN LE, GUS SCHRADER, AND ALEXANDER SHAPIRO
Figure 2.
At left: triangulation of D . At right: triangulation of the punc-tured digon D ˝ .labeling. We denote the union of all regions with label G by S G , and that of all regions withlabel T by S T . Definition 1.2. A G -gate (resp. T -gate) is an interval in S G X B S (resp. S T X B S ). A set G of disjoint gates in S will be called a gating of S . In the special case that G consists of aunique gate in each T -region, we will say S is notched . Definition 1.3. An n -gon D n is a G -disk decorated with n contractible T -regions along theboundary (see Fig. 2). Definition 1.4. A triangulation ∆ is a presentation of S as a union of triangles D , suchthat the intersection of any pair of triangles is either empty or a union of (one, or two)digons D , and the intersection of any triple of triangles is empty. We shall often refer to thedigons in a triangulation of a decorated surface as the edges of the triangulation. A notchedtriangulation of a notched S is such that each end of each digon lies in the unique T -gate. Remark 1.5.
A decorated surface S is called simple if it admits a triangulation. In a simpledecorated surface, each T -region is either a disk contracting to a point of B S or an annuluscontracting onto an entire component of B S . We call such T -regions marked points and punctures respectively. Isotopy classes of simple decorated surfaces are in bijection withmarked surfaces appearing in [FG06], while isotopy classes of triangulations of decoratedsurfaces are in bijection with those of ideal triangulations – maximal collections of non-intersecting arcs with endpoints at marked points or punctures – of the corresponding markedsurfaces S . A notched triangulation determines a notched ideal triangulation – an idealtriangulation with a distinguished angle (see Fig. 4). Definition 1.6.
Let S be a decorated surface. A decorated local system E on S is a triple E “ p E G , E T , E B q , consisting of a G -local system E G on S G , a T -local system E T on S T , anda reduction E B to B of the product G ˆ T -local system E G ˆ E T over C .Recall that a G -local system means a principal G -bundle with flat connection. A reductionof a local system to a subgroup H means the specification of a flat H -subbundle. We shallconsider the group embedding B ã Ñ G ˆ T obtained by composing the diagonal embedding of B into B ˆ B with the inclusion of B in G and the projection to T . In this case, a B -reductionamounts to specifying an element of the coset space B zp G ˆ T q – N z G , satisfying a certaincompatibility with the G and T -monodromies near the curve C (see Example 1.10). Definition 1.7.
The decorated character stack Ch p S q is the moduli stack of decoratedlocal systems on S . UANTUM DECORATED CHARACTER STACKS 5 p p A q ( P ) D T p A q p P q Figure 3.
Gated decorated surfaces corresponding to moduli problems at apuncture are depicted. See Remark 1.20 for explanation of the ( A ) case.Now fix a gating G of S . Then by a framing of E we will mean a trivialization of the localsystems E G and E T at each G - and T -gate, respectively. Definition 1.8.
The framed decorated character stack Ch fr G p S q is the moduli stack ofpairs, consisting of a decorated local system, and a framing of the local system over G .Suppose that G consists of at least one gate in every region of S . In this case, the modulistack is in fact a variety, since an isomorphism E Ñ E of decorated local systems is uniquelydetermined by the condition that it preserve trivializations at each gate. This allows us topresent the decorated character stack concretely as the stack quotient:Ch p S q – Ch fr G p S q{p G m ˆ T n q , of an affine variety by a reductive algebraic group, where G m ˆ T n acts by changing framingsat the m G -gates and n T -gates of G . Example 1.9.
Consider the n -gon D n . The only G - and T -local systems are the trivial one,hence the only data are the B -reductions, given by elements of B zp G ˆ T q – N z G . Hence,putting a G -gate in the G -region and a T -gate in each T -region, we have:Ch fr G p D n q “ p N z G q ˆ n and Ch p D n q “ T n zp N z G q ˆ n { G. Example 1.10.
Let us consider the four decorated and gated surfaces depicted in Fig. 3,and their decorated character stacks. In each one, the G -local system E G is prescribed byan element g of G recording the monodromy in the G -region. For p ˆ A q and p P q , the T -localsystem E T is given by an element h of T , while for ( A ) and ( P ) we must have h “ e since E T must be trivial. Finally, in each case we need to specify a principal B -sub-bundle of therestriction of E G ˆ E T over the wall. Such a sub-bundle is specified by a Borel subgroup B containing g . To summarize, in each case Ch fr G p S q parameterizes:( ˆ A ) An element r F P N z G at the puncture, with a requirement that the monodromy in the G -region preserves π p r F q P B z G .( P ) An element F P B z G at the puncture, with a requirement that the monodromy in the G -region preserves F .( A ) An element r F P N z G at the puncture, with a requirement that the monodromy in the G -region fixes r F .( P ) An element F P B z G at the puncture, with a requirement that the monodromy in the G -region preserves F , and is unipotent. Hence our use of ‘framing’ differs from [FG06], where it refers to a B -reduction rather than a trivialization. DAVID JORDAN, IAN LE, GUS SCHRADER, AND ALEXANDER SHAPIRO
The p ˆ A q space serves as our main workhorse in this paper, because it controls the othersby regulation of T -framings. The p A q space is the preimage of the identity with respectto a multiplicative moment map µ valued in T . The p P q space is the quotient of the p ˆ A q space by the T -action changing the framing. Accordingly, the p P q space is the multiplicativeHamiltonian reduction of the ˆ A space. These inter-relations are summarized in the diagrambelow p ˆ A qp A q p P qp P q Ð Ñ ´{ T ÐÑ µ ´ p e q{ T Ð â Ñ µ ´ p e q Ð Ñ ´{ T Ð â Ñ µ ´ p e q Remark 1.11.
For simplicity, we have considered only a single puncture and no markedpoints in the above discussion. In general, each puncture contributes one datum of each type,while each marked point contributes a factor of N z G , with no condition. There are also ( X )and p X q spaces, which have ( P ) data at each puncture, but instead have a B z G factor foreach marked point. These are obtained from p P q spaces by quotienting the T -action at eachmarked point.1.2. Construction via factorization homology.
Our starting point is the following fun-damental result, which is a direct corollary of Theorem [BFN10, Theorem 1.2]:
Theorem 1.12.
For any n -manifold X , we have an equivalence of categories, QC p Ch p X qq “ ż X Rep p G q . On the left we denote the category of quasi-coherent sheaves on the (ordinary, not decorated)character stack. The integral notation on the righthand side denotes the factorization homologyof X with coefficients in the symmetric monoidal category Rep p G q , which is here regarded asan E n -algebra. The theorem reconstructs the character stack Ch p X q on any n -manifold X interms of its factorization homology.We may treat decorated character stacks analogously in the framework of stratified factor-ization homology . Whereas an ordinary surface has only a single basic open disk (to which isattached Rep p G q above), a decorated surface has three basic open disks: one contained in S G ,one contained in S T , and one straddling a wall.Accordingly, factorization homology of decorated surfaces requires us to specify coefficientsfor each type of basic open disk, together with a D isk -algebra — a system of functors andhigher coherences relating the coefficients. For instance, in the undecorated case we wererequired to specify not only Rep p G q , but also its E (= braided monoidal) structure. Inorder to recover decorated character stacks, we will specify the so-called “parabolic inductionalgebra”, being the triple of categoriesRep p T Ð B Ñ G q : “ p Rep p G q , Rep p B q , Rep p T qq , together with the pullback functors along i : B Ñ G and π : B Ñ T . The following theoremagain follows as a corollary of [BFN10, Theorem 1.2]: UANTUM DECORATED CHARACTER STACKS 7
Theorem 1.13.
We have an equivalence of categories, QC p Ch p S qq “ ż S Rep p T Ð B Ñ G q . In the works [BBJ18a; BBJ18b], Theorem 1.12 was taken as the starting place for a quanti-zation procedure.
Namely, replacing Rep p G q with Rep q p G q , yields a functorial deformationquantization of Ch p S q . Such quantizations of character stacks via factorization homology weresubsequently related to combinatorial Chern–Simons theory and Alekseev–Grosse–Schomerusalgebras [AGS96] in [BBJ18a], double affine Hecke algebras [Che05] in [BBJ18b], and to skeinalgebras and skein categories [Tur91; Prz91] in [Coo19].In the present paper we extend this paradigm to the study of decorated surfaces by replacingRep p T Ð B Ñ G q with the “quantum parabolic induction algebra”. This consists of thetriple, Rep q p T Ð B Ñ G q : “ ` Rep q p G q , Rep q p B q , Rep q p T q ˘ along with the ribbon braided tensor structure on Rep q p G q and Rep q p T q , the tensor structureon Rep q p B q , and a braided tensor functorRep q p G q b Rep q p T q ÝÑ Z p Rep q p B qq , where Z p Rep q p B qq denotes the Drinfeld center of the monoidal category Rep q p B q , andRep q p T q carries the opposite of its standard braiding. This data defines a local coefficientsystem for stratified factorization homology of [AFT17], see also [BJS18, Section 3]. Hence,we make the following Definition 1.14.
The quantum decorated character stack Z p S q is the stratified factor-ization homology, Z p S q : “ ż S Rep q p T Ð B Ñ G q . The two most important features of stratified factorization homology are that it is functorialfor stratified embeddings, and satisfies the excision property. To recall the excision property,we first recall that any cylindrical decorated X carries a monoidal structure by stacking in thecylinder direction, and that any embedding of X into a neighborhood of the boundary of S induces a Z p X q -module category structure on Z p S q .Excision computes the stratified factorization homology of two decorated surfaces S and S glued over a cylindrical decorated surface S as a relative tensor product Z p S q “ Z p S q b Z p S q Z p S q . For example, Fig. 1 shows the equivalence Z p D q » Rep q B b Rep q G Rep q B b Rep q G Rep q B, while Fig. 2 illustrates Z p D q » Z p D q b Z p D q Z p D q and Z p D ˝ q » Z p D q b Z p D q b Z p D q Z p D q . Notation 1.15.
Given a set G of m G -gates and n T -gates, we will use the followingabbreviations where convenient.Rep q p G q “ Rep q p G q m b Rep q p T q n “ Rep q p G m ˆ T n q . DAVID JORDAN, IAN LE, GUS SCHRADER, AND ALEXANDER SHAPIRO
We refer to the action of Rep q p G q on Z p S q induced by G ˆ I ã Ñ S as disk insertion . Toimprove readability throughout the paper, we will write G , T , G ˆ T , etc. in place of G foremphasis as needed, and we will abbreviate “Rep q p G q -module category” and “Rep q p G q -modulefunctor” by G -module category, G -module functor, respectively.A final important ingredient of stratified factorization homology is the existence of acanonical distinguished object Dist S in each category Z p S q . The assignment S ÞÑ Z p S q isfunctorial for stratified embeddings, and the morphism in factorization homology inducedby the empty embedding H ã Ñ S determines a functor Vect Ñ Z p S q ; its image on theone-dimensional vector space is, by definition, the distinguished object. Classically, thedistinguished object is precisely the structure sheaf of Ch p S q , hence the distinguished objectquantizes the structure sheaf to Z p S q , and the functor Hom p Dist S , ´q quantizes the globalsections functor.1.3. Aside on noncommutative algebraic geometry.
Our main results are phrased inthe language of noncommutative algebraic geometry, for which we follow [Smi00]. The mainidea is to treat an arbitrary abelian category C , such as Z p S q , as the category of sheaveson some putative noncommutative algebraic variety (more generally, stack), and to study C functorially drawing on geometric intuition coming from the commutative case. Hence, eachdefinition below, when applied in the commutative case q “
1, will recover precisely the usualgeometric notion, and for generic q will give its deformation.In this context, a full subcategory i : C Ñ D is called open if the inclusion functor i admits an exact left adjoint. The name “open” comes from the following basic example:given an inclusion i : U ã Ñ X of an open subvariety (more generally, an open substack), thepushforward i ˚ : QC p U q Ñ QC p X q is fully faithful with exact left adjoint i ˚ . For this reason,we will refer to i L as the restriction functor , and talk about restricting objects to opensubcategories. Important examples of open subcategories arise from Ore localizations: theinclusion A r S ´ s´ mod Ă A ´ mod has an exact left adjoint — the localization functor — givenby tensoring over A with A r S ´ s .Dually, we may view an open subcategory C as a quotient of D . Indeed ker p i L q is a Serresubcategory, hence i L induces an equivalence C » D { ker p i L q . Classically, ker p i ˚ q consists ofthe sheaves supported on the complement of U . The geometric union of open subcategories i : C Ñ D and i : C Ñ D is defined as C Y C “ D {p ker i X ker i q . We emphasize that thisis not simply the full subcategory obtained by taking unions of their objects.We say that an open subcategory C Ă D is a chart (alternatively, is affine ) if C admitsa compact projective generator X , and hence an equivalence C » End p X q op ´ mod with thecategory of modules for a (typically, noncommutative) algebra. Classically, this would meanthat the corresponding open set is affine. We say that the chart is toric if, moreover, thealgebra End p X q op is a quantum torus, i.e. if there exists an isomorphism of algebras,End p X q op » C @ x ˘ , . . . , x ˘ m D { x x i x j “ q m ij x j x i y , for some skew symmetric integer matrix M “ p m ij q .Given two charts C and C of D , the transition functor φ C , C : C Ñ C is the compositeexact functor, φ C , C : C i C ÝÑ D i L C ÝÑ C , where i C , i C are inclusion functors, and i L C is left adjoint to i C . In the case C “ A ´ mod and C “ A ´ mod are affine charts, the transition functor φ C , C : C Ñ C is given by tensoring withsome A ´ A bimodule M . Classically, if C “ QC p U q and C “ QC p U q with U, U being open UANTUM DECORATED CHARACTER STACKS 9 affine subvarieties of some variety X , the bimodule M is merely the algebra of functions onthe intersection U X U .An additional feature in our story is the presence of quantum categorical G - and T -actionscoming from disk insertions through G and T -gates. We will say that a subcategory is a G -chart (alternatively, is G -affine ) if it admits a compact projective Rep q p G q -generator X ,hence an equivalence, C » End G p X q op ´ mod G , where the notation A ´ mod G is a shorthand for the category of A -modules internal to Rep q p G q – i.e. the category of A -modules equipped with compatible actions of the quantum groupattached to G .Classically for a quotient stack X { G , QC p X q being G -affine means that the map X Ñ ‚{ G isaffine. For instance ‚{ G itself is not affine, since QC p‚{ G q “ Rep p G q does not admit a compactprojective generator; however it is G -affine, since the trivial representation is tautologically acompact projective G -generator. See Section 3 for a discussion of monadic reconstruction andinternal endomorphism algebras, and Section 3.3 for discussion of familiar classical examplesin this language.1.4. Charts and flips on Z p S q . As explained in Section 2, the framework of stratifiedfactorization homology may be understood as the specification of a universal property to besatisfied by an desired invariant of surfaces, and a theorem stating that this universal propertycan be solved in suitable settings. The universal property itself is phrased in -categoricalterms, and its a priori solution is expressed via colimits in higher categories.At first encounter, such a definition involving -colimits may sound hopelessly abstract.However, a number of techniques — most notably the excision formula, and its monadicreformulations — allow us to wrangle this abstract definition into a concrete enough formfrom which we can extract the calculus of quantum cluster algebras. Our strategy is roughlyas follows; we will focus the exposition on G “ SL , however, we will explain the modificationsfor the PGL -case as well.To begin, we treat the case of the n -gon D n , with a unique gate in the G -region and in each T -region. We compute the internal endomorphism algebra F q p Conf frn q “
End G ˆ T n p Dist S q ,identifying it with the braided tensor product of n copies of an algebra F q p N z G q . The functorHom G ˆ T n p Dist S , ´q , of G ˆ T n -equivariant global sections, defines an open embedding, i : Z p D n q ã Ñ F q p Conf frn q´ mod G ˆ T n In Proposition 3.51 we compute the orthogonal of Z p D n q to be the Serre subcategory of torsion modules (see Definition 3.42).We consider triangulations ∆ of D n . We define U q p g q -invariant Ore sets S ∆ in F q p Conf frn q ,whose generators correspond to edges of ∆, and which quantize corresponding cluster A -coordinates on G zp N z G q ˆ n . In fact, the most important cases are n “ ,
3, which each have aunique triangulation. We denote these special triangulations by D (a degenerate triangulationwith one edge, but no triangles) and D , respectively. For n “ S D consists of monomialsin a single element denoted D , and for n “ S D consists of monomials in elements denoted D , D , D , and coming from the three inclusions of D as an edge of D .In Section 3.8, we define an open subcategory Z p ∆ q Ă Z p D n q , by the condition that theelements of S ∆ act invertibly. We show that the restriction Dist ∆ “ Res Z p ∆ q p Dist D n q of thedistinguished object to Z p ∆ q is a G ˆ T n -compact-projective generator (we note that it is not so for Z p D n q ), hence yielding an equivalence of categories Z p ∆ q » F q p Conf frn qr S ´ s´ mod G ˆ T n , and sandwiching Z p D n q between G ˆ T n -affine categories: F q p Conf frn qr S ´ s´ mod G ˆ T n Ă Z p D n q Ă F q p Conf frn q´ mod G ˆ T n , Crucially, we also show that the functor of taking G -invariants is conservative on Z p ∆ q (wenote again, it is not so for Z p D n q ), hence we obtain an equivalence of categories, Z p ∆ q » End T n p Dist ∆ q´ mod T n – ´ F q p Conf frn qr S ´ s ¯ U q p g q ´ mod T n . Henceforth let us abbreviate ζ p ∆ q : “ End T n p Dist ∆ q´ mod T n . Returning to the special cases n “ ,
3, we calculate isomorphisms, ζ p D q – C q @ D ˘ D and ζ p D q – C q @ D ˘ , D ˘ , D ˘ D { I, where C q “ C p q ˘ { q , and I consists of the q-commutation relations, D D “ q ´ D D , D D “ q ´ D D , D D “ q ´ D D . We then turn to the construction of charts on a general simple decorated surface Z p S q ,for which we appeal to excision. In simple cases like D n , one might imagine building anarbitrary surface inductively by gluing triangles of a triangulation in one by one. To givea more uniform formula and to allow for self-gluings, however, we elect instead to glue alltriangles together in one go, by writing S “ p D q \ t ğ p D q \ (cid:96) p D q \ (cid:96) . (1.1)Excision gives an equivalence of categories, Z p S q » Z p D q t ò Z p D q (cid:96) Z p D q (cid:96) , (1.2)Hence we define the open subcategory Z p ∆ q Ă Z p S q by gluing the open subcategories fortriangles over digons, accordingly: Z p ∆ q : “ Z p D q t ò Z p D q (cid:96) Z p D q (cid:96) , (1.3) Remark 1.16.
The reader is encouraged to view Eq. (1.2) in analogy with the formula forHochschild homology of an associative algebra A with coefficients in its bimodule M , HH p A, M q “ A b A b A op M. (1.4)In this analogy, A is the monoidal category obtained from the union of all the digons, M isthe bimodule category obtained as the unions of all the triangles, and the bimodule structurecomes from the two inclusions of each edge into the triangles it borders.Having defined the charts by gluing, we proceed to computing their global sections. Weuse the following particular case of [BBJ18a, Theorem 4.12]: Lemma 1.17.
Let C be a braided monoidal category with a braided commutative algebra object A P C , and a pair M , N , of A -algebras in C . Then we have an equivalence of categories, M ´ mod C b A ´ mod C N ´ mod C » p M b A N op q ´ mod C . UANTUM DECORATED CHARACTER STACKS 11
However, in order to apply Lemma 1.17, we turn out to require an intermediate step ofopening additional T -gates to ensure that the algebras we wish to tensor over is in fact braidedcommutative. This being done, we can glue using Lemma 1.17, and finally close all butone gate in each T -region to obtain our main results. By opening a gate , we refer to theoperation of pulling the T -action on Z p S q back through the tensor functor, b : Rep q p T q Ñ Rep q p T q . In order to close a gate , we apply the functor of taking T -invariants at the gate.Now, let us open two gates in each T -region of D and consider the category Z p D q asa T -module category. The result is still T -affine with the T -progenerator Dist D , andin Lemma 4.7 we calculate the algebra, r ζ p D q “ End T ´ Dist D ¯ ´ mod T . It is again a quantum torus, now with six generators. We have equivalences of categories Z p D q » r ζ p D q´ mod T » ζ p D q´ mod T . We similarly define an algebra r ζ p D q by opening a pair of T -gates in each T -region of a digon D , and have equivalences Z p D q » r ζ p D q´ mod T » ζ p D q´ mod T . Thanks to Lemma 1.17, we conclude that the chart Z p ∆ q is toric and G -affine , that is Z p ∆ q » r ζ p ∆ q´ mod T t for a quantum torus algebra r ζ p ∆ q obtained as a braided tensor product as in Lemma 1.17.We provide an explicit presentation of r ζ p ∆ q in Proposition 4.11.We now choose a distinguished gate at every T -region of S , and close all but the distinguished T -gate. This fixes notchings of S and ∆; we denote the altter by ∆ ‚ . We obtain a quantumtorus ζ p ∆ ‚ q from r ζ p ∆ q , as the subalgebra of invariants with respect to the T -actions at allnon-distinguished T -gates.The quantum torus ζ p ∆ ‚ q has a simple presentation, which depends only on the isotopyclass of ∆ ‚ . There is one generator Z e for each edge e of the notched triangulation, and anadditional generator α v for each puncture. The q -commutation relations between generatorsare described by simple formulas depending on the incidence relations and total ordering ateach vertex, see Theorem 4.16. Remark 1.18.
We note that D n has no punctures, and the resulting algebra ζ p ∆ ‚ q doesindeed coincide with ζ p ∆ q computed above as a subalgebra of F q p Conf frn qr S ´ s .Throughout the discussion above, we have fixed G “ SL . In fact, our computationsimmediately give rise to charts Z PGL p ∆ q Ă Z PGL p S q as well, with the correspondingquantum tori ζ PGL p ∆ ‚ q being realized naturally as subalgebras of ζ SL p ∆ ‚ q (see Corollary 4.8and Proposition 4.20). Theorem 1.19.
Let S be a simple decorated surface with k T -regions, with gating G “ T k consisting of a single gate in each T -region. Let G be either SL or PGL .(1) Each isotopy class of triangulations ∆ determines a G -toric chart Z p ∆ q Ă Z G p S q . Theadditional choice of the isotopy class of a notched triangulation ∆ ‚ defines a quantumtorus ζ G p ∆ ‚ q , and equivalence of categories: Z G p ∆ q » ζ G p ∆ ‚ q´ mod T k . (2) Given two notched triangulations ∆ ‚ and ∆ differing by a flip of a single edge, thefunctor Z G p ∆ q Ñ Z G p ∆ q is induced by an explicit birational isomorphism ζ p ∆ ‚ , ∆ q between ζ G p ∆ ‚ q and ζ G p ∆ q described in Proposition 4.25 for the case G “ SL , andby its restriction in the case PGL . Remark 1.20.
Consider the cluster K -variety structure on the moduli space of twistedSL -local systems on S , see [FG06] and [FST08]. Let A ∆ be a cluster chart arising from theideal triangulation ∆. If the surface S has only boundary marked points and no punctures,the cluster algebra A ∆ admits quantization in the sense of [BZ05], see [Mul16]. In that case,we denote the quantum chart arising from the triangulation ∆ by A q ∆ . Then the followingobservations are immediate.(1) If S has no punctures, ζ p ∆ q – A q ∆ , and the flip between ζ p ∆ ‚ q and ζ p ∆ q coincideswith the corresponding quantum cluster mutation.(2) If S has punctures, the elements α v are not central in ζ p ∆ ‚ q . However, upon setting q “
1, the algebra ζ q “ p ∆ ‚ q becomes commutative and one may consider an ideal I Ă ζ q “ p ∆ ‚ q generated by all elements of the form α v ´
1. Then, ζ q “ p ∆ ‚ q{ I – A ∆ ,and the flip between ζ p ∆ ‚ q and ζ p ∆ q descends to the cluster mutation between thecharts A ∆ and A ∆ .In this way, one recovers all cluster charts which are labelled by triangulations without verticesincident to two or more tagged arcs; we refer the reader to [FST08] for the definition of atagged arc.Now, let S be a simple decorated surface with p punctures and m marked points, with asingle gate in each T -region. Let χ p ∆ q “ ζ p ∆ ‚ q T p be the subalgebra of invariants with respectto the T -action at the punctures. Note that χ p ∆ q is still a quantum torus and depends onlyon the triangulation ∆ rather than on the notched triangulation ∆ ‚ . In this way, we can givean alternative formulation of Theorem 1.19 using the χ p ∆ q -tori instead of ζ p ∆ ‚ q .Before proceeding to this formulation in Theorem 1.21, let us briefly discuss the case G “ PGL , and its relation to G “ SL . Recall that the algebra r ζ SL p ∆ q carries a T t -action,where t is the number of triangles in ∆. This action endows r ζ SL p ∆ q with a Z t -grading.Then r ζ PGL p ∆ q coincides with the subalgebra of r ζ SL p ∆ q , generated by elements of evendegree with respect to the T -action at any T -gate. Closing all gates but the distinguishedgate in each T -annulus, we obtain an embedding ζ PGL p ∆ ‚ q Ă ζ SL p ∆ ‚ q . Furthermore, let χ PGL p ∆ q Ă ζ PGL p ∆ ‚ q denote the subalgebra of invariants with respect to the T -actionat punctures as described previously in the SL case. We have the following commutativediagram: χ SL p ∆ q ζ SL p ∆ ‚ q χ PGL p ∆ q ζ PGL p ∆ ‚ qÐ â Ñ Ð â Ñ Ð â Ñ Ð â Ñ We describe the generators of the subalgebra χ SL p ∆ q Ă ζ SL p ∆ ‚ q in Section 4.5, where weshow that χ SL p ∆ q is a free module over χ PGL p ∆ q of rank 2 N , where N “ dim H p S, M q isthe dimension of the first homology of S relative to the subset M of marked points.Finally, let us recall from [FG06; FG09a], the quantum X -variety structure on the modulispace of PGL -local systems with pinnings on a marked surface S . Let X q ∆ denote the quantumcluster chart arising from the triangulation ∆. Then we obtain the following result. UANTUM DECORATED CHARACTER STACKS 13
Theorem 1.21.
Let S be a simple decorated surface with m singly gated marked points. Leteither G “ SL and m ě , or G “ PGL and m ě .(1) The isotopy class of each triangulation ∆ of S determines a quantum toric chart Z G p ∆ q Ă Z G p S q with quantum torus χ G p ∆ q and an equivalence of categories: Z G p ∆ q » χ G p ∆ q´ mod T m . (2) Given two such triangulations ∆ ‚ and ∆ differing by a single flip, the functor Z G p ∆ q Ñ Z G p ∆ q is the birational isomorphism obtained by restricting the ˆ A -flip ζ p ∆ ‚ , ∆ q between ζ p ∆ ‚ q and ζ p ∆ q described in Proposition 4.25 to the subalgebras χ p ∆ q , χ p ∆ q .(3) We have isomorphisms χ PGL p ∆ q – X q ∆ , and the flip between χ PGL p ∆ q and χ PGL p ∆ q coincides with the quantum cluster mutation of X -variables between X q ∆ and X q ∆ . Remark 1.22.
One may also consider the subalgebra, η G p ∆ q : “ χ G p ∆ q T m Ă χ G p ∆ q , of invariants with respect to the remaining T -actions. For G “ PGL , one obtains anequivalence of categories Z G p ∆ q » η G p ∆ q´ mod , and the quantum tori η G p ∆ q coincide with quantum cluster charts on the X -variety consideredin [FG06]. In the language of cluster algebras, one obtains the quiver for η G p ∆ q from the onefor χ G p ∆ q by erasing all of the frozen nodes.In the case G “ SL , some additional care is needed in closing the final gate, owing to thenon-trivial center of SL . See Remark 4.19 for more details. Remark 1.23.
Setting q “
1, the homomorphism χ q “ p ∆ q ã Ñ ζ q “ p ∆ ‚ q{ I described abovecoincides with the cluster ensemble map between the X - and A -cluster structures discussedabove. In case S has no punctures, the homomorphism χ PGL p ∆ q ã Ñ ζ SL p ∆ ‚ q in Theorem 1.21coincides with the quantum cluster ensemble map. In particular our construction may beregarded as the extension of the cluster ensemble map, to surfaces with punctures and to q ‰ A -varieties in previous remarks, we have set q “
1. This is necessarymerely because the ideal I is only a one-sided ideal for q ‰
1, which is in turn a manifestationof the fact that in the presence of punctures, A -varieties are not naturally Poisson. Thepromotion to ˆ A -varieties by allowing non-trivial monodromy of the T -connection has preciselyfixed this issue, by adding a new generator for each puncture. Classically ˆ A varieties arePoisson (and in fact symplectic in the absence of marked points), and we have constructedtheir deformation quantizations.Still, one might wonder if quantum A -varieties nevertheless appear directly in factorizationhomology terms. Topologically, the picture to consider is the disk labelled p A q in Fig. 3, whichwe will denote here by D ‚ ; we will also denote by D ˝ the disk labelled p ˆ A q . We can includethe disk D T into the filled inner T -region of D ‚ as depicted in the figure, and thereby obtain afunctor Rep q p T q Ñ Z p D ‚ q . However, since D T does not enter D ‚ through a gate, this functoris not a T -module functor. We therefore can construct End T p Dist D ‚ q formally as we do in thepresence of a gate, however owing to this lack of module structure, it is only a plain object –namely End T p Dist D ˝ q{x α p ´ y P Rep q p T q – and not an algebra object. a b cdα a b cd α a b cd α Figure 4.
Notched triangulations (above), corresponding to notched idealtriangulations (below), of the punctured digon D ˝ . Example 1.24.
Consider a punctured digon D ˝ . Three isotopy classes of its notched tri-angulations are shown in the top row of Fig. 4, the bottom row of the same figure showsthe corresponding notched ideal triangulations. Let ζ SL : “ ζ SL p ∆ l ‚ q , where ∆ l ‚ is thetriangulation on the left. Then ζ SL “ C r q ˘ s @ Z ˘ a , Z ˘ b , Z ˘ c , Z ˘ d , α ˘ D with the following relations: Z a Z b “ q ´ Z b Z a , Z b Z c “ q ´ Z c Z b , αZ a “ Z a α,Z a Z c “ Z c Z a , Z b Z d “ q ´ Z d Z b , αZ b “ qZ b α,Z a Z d “ q Z d Z a , Z c Z d “ q ´ Z d Z c αZ c “ Z c α,αZ d “ qZ d α. Let us also set ζ c SL : “ ζ SL p ∆ c ‚ q and ζ r SL : “ ζ SL p ∆ r ‚ q , where ∆ c ‚ and ∆ r ‚ are the triangula-tions in the left and on the right of Fig. 4. Then the quantum tori ζ c SL and ζ r SL are obtainedfrom ζ SL by replacing the generator Z d with Z d and Z d respectively. Then Z d “ q ´ Z d α ´ , while the flip between ζ SL and ζ c SL reads Z d “ : Z ´ d αZ a Z b : ` : Z ´ d Z c Z b : “ q ´ Z ´ d αZ a Z b ` q Z ´ d Z c Z b , where : : denotes the Weyl ordering introduced in Definition 4.14, and we obtain relations Z c Z d “ qZ d Z c , Z a Z d “ q ´ Z d Z a , Z b Z d “ Z d Z b , αZ d “ Z d α. Let us now focus on the triangulation ∆ l ‚ , and describe the quantum subtori of ζ SL obtainedfrom the latter by taking T -invariants at the puncture or marked points, and by replacing UANTUM DECORATED CHARACTER STACKS 15 SL with PGL . We have ζ PGL “ C “ q ˘ ‰ @ Z ˘ a , Z ˘ b , α ˘ , p Z a Z b Z d q ˘ , p Z b Z c Z d α q ˘ D Ă ζ SL and χ SL “ C “ q ˘ ‰ @ Z ˘ a , Z ˘ c , α ˘ , p Z b Z ´ d q ˘ D Ă ζ SL . Then, setting X a “ : Z a Z ´ b Z d : , X b “ : αZ a Z ´ c : , X c “ : α ´ Z c Z ´ d Z b : , X d “ : αZ c Z ´ a : , we obtain χ PGL “ C “ q ˘ ‰ @ X ˘ a , X ˘ b , X ˘ c , X ˘ d D “ ζ PGL X χ SL . Finally, we also have η SL “ C “ q ˘ ‰ @ p Z a Z ´ c q ˘ , α ˘ D Ă χ SL ,η PGL “ C “ q ˘ ‰ @ X ˘ b , X ˘ d D Ă χ PGL . We end this example with the following observation. In [SS19], the third and fourth authorsshowed that the quantum group U q p sl n q is closely related to the quantum cluster Poissonvariety on the P -space for G “ P GL n . Let us recall this relation in the setup of the presentpaper. In case n “
2, the Drinfeld double D of the quantum Borel U q p b q Ă U q p sl q is thealgebra D “ C p q q @ e, f, k ˘ ω , k ˘ ω D with relations k ω e “ qek ω , k ω e “ q ´ ek ω , k ω k ω “ k ω k ω ,k ω f “ q ´ f k ω , k ω f “ qf k ω , and r e, f s “ p q ´ ´ q qp k α ´ k α q , where k α “ k ω and k α “ k ω . Here we denote by ω the fundamental weight of sl , and by α “ ω the positive root. Then one has an injective homomorphism ι : D ã Ñ χ SL given by e ÞÝÑ Z b , f ÞÝÑ Z d , k ω ÞÝÑ Z a , k ω ÞÝÑ Z c . Distinguished objects and global sections.
The charts constructed in Theorem 1.19and Theorem 1.21 are not arbitrary: the required generator of each chart is the restriction ofthe distinguished object. We may therefore consider the restriction Dist U “ Res U p Dist S q ofthe distinguished object to the subcategory U “ Y Z p ∆ q covered by all open charts Z p ∆ q .As a corollary of Theorem 1.19, we have that End T k p Dist U q is isomorphic to the subalgebraof ζ p ∆ ‚ q consisting of those elements which remain regular, that is Laurent polynomials, underevery quantum flip (clearly, this algebra is independent of the triangulation ∆ ‚ ). Similarly,as a corollary of Theorem 1.21, we have that End T d p Dist U q and End p Dist U q are isomorphicrespectively to the subalgebras of χ p ∆ q and η p ∆ q , consisting of elements regular under allflips.Now, let L p P S q and L p X S q denote the universally Laurent algebras from the cluster Poissonstructure [FG06; GS19] on the P - and X -moduli space respectively. The algebras L p P S q and L p X S q consist of elements that are Laurent polynomials in the cluster coordinates ofevery chart. We denote by L q p P S q and L q p X S q the quantum versions [FG09a; FG09b] of In [SS19], the above homomorphism is written in terms of E “ i k ω ek ´ ω , F “ i k ω fk ´ ω , K “ k α , and K “ k α ,where i “ ?´ D , generated by E, F, K, K , is mappedinto χ PGL Ă χ SL under ι . these algebras. Similarly, let L p A S q be the universally Laurent algebra, also known as theupper cluster algebra, from the cluster K -structure on the moduli space of twisted SL -localsystems on S , considered in [FG06]. If S has no punctures, we denote by L q p A S q the quantizedversion [BZ05] of L p A S q . Then we have the following Corollary, which follows from the factthat the tori ζ ∆ ‚ and flips between them coincide with the quantum cluster charts on L q p A S q and their mutations. Corollary 1.25. If S has no punctures and G “ SL , we have the following isomorphism ofalgebras in Rep q p T k q : End T k p Dist U q » L q p A S q . We expect this description to extend to the following cases:(1) For G “ SL , we have the following isomorphism of commutative rings:End T k p Dist U q| q “ { I » L p A S q . (2) For G “ P GL , there is an isomorphisms of algebras in Rep q p T d q End T d p Dist U q » L q p P S q . (3) For G “ P GL , there is an isomorphisms of algebrasEnd p Dist U q » L q p X S q . The only reason the above equivalences are non-obvious as stated is that we have notincluded certain special charts in our definition of U , here we call a chart special if it containsa vertex incident to two or more tagged arcs. Hence to prove the identities, it would suffice toshow that an element which is regular in every non-special (quantum) cluster chart, is in factregular in every (quantum) cluster chart, or alternatively we could incorporate special chartsinto the definition of U .A more interesting question is whether the above identities hold with Dist S in place of Dist U .Classically, this is the question whether U c has codimension at least two in the decoratedcharacter stack. We intend to return to such questions in a future paper.1.6. Outlook.
The most self-evident challenge which presents itself is to extend the resultsof this paper to groups of higher rank, namely to discover cluster charts and flips withinquantum decorated character stacks for higher rank groups. While the general constructionfrom this paper applies to arbitrary reductive groups, we expect the detailed determination ofcluster charts beyond rank one will involve many interesting and challenging aspects, and willnot be a straightforward modification of our computations here. In higher rank, decoratedcharacter stacks admit richer algebraic structure — an arbitrary parabolic subgroup P Ă G with Levi quotient P Ñ L determining a domain wall between G and L . These domain wallscan not only stack (i.e, compose as 1-morphisms in BrTens), but they may also overlap —this is modelled by the 2-morphisms in BrTens, and hence involves stratified surfaces withpoint defects. We expect a correspondence between these interesting additional structures onthe lattice of Levi subgroups, and the rich combinatorial structure of the cluster charts inhigher rank.Let us mention a number of more immediate applications which we intend to pursue: ‚ Combining our work with that of [Coo19] gives rise to embeddings of skein algebras intoquantum cluster algebras. These should be related to — and extend — the well-knownconstructions of [Mul16] and [BW11]. We will give an invariant construction of suchembeddings in a forthcoming work.
UANTUM DECORATED CHARACTER STACKS 17 ‚ Alekseev–Grosse–Schomerus have obtained [AGS96] a generators-and-relations pre-sentation of certain algebras, now commonly called AGS algebras, by quantizing thePoisson structure described in [FR99] by Fock and Rosly on framed character varieties(in the absence of T -regions). These algebras are now understood via monadic recon-struction for factorization homology, in [BBJ18a], and subsequently in terms of stated [CL19] and internal [GJS19] skein algebras. The techniques in this paper thereforelead us to quantum cluster embeddings of arbitrary AGS algebras, the simplest caseof which is discussed already in Example 1.24. We will explore this systematically ina forthcoming work. ‚ While this paper focuses primarily on construction of the subcategory Y Z p ∆ q and itsrelation to quantum cluster algebras, an important direction of future inquiry is tounderstand precisely the complement , which describes the stacky points of decoratedcharacter stacks and their quantizations. ‚ By appealing to the 4-category BrTens, we may regard our constructions as giving the2-dimensional part of a fully extended p ` (cid:15) q -dimensional TFT. In particular, we mayconsider extension of our work to decorated 3-manifolds, for example, to hyperbolicknot complements. We expect close relations to [Dim13; DGG16]. ‚ We have largely restricted attention to simple decorated surfaces in the present work.While there is no reason to expect something like cluster charts for decorated surfaceswhich are not simple, we nevertheless expect that many interesting examples will comefrom studying non-simple quantum decorated character stacks, with connections toHecke categories, quantum Beilinson-Bernstein theorem [BK06; Tan05], dynamicalquantum groups, and numerous other topics in quantum representation theory whereparabolic induction plays a role.
Outline.
Let us now outline of the contents of this paper. In Section 2, we recall basicnotions from stratified factorization homology and representations of quantum groups, leadingto the formal definition of Z p S q . In Section 3, we review monadic reconstruction, do numerouscomputations with it, and finally use it to construct cluster charts on Z p D n q . In Section 4,extend the construction to general S , and discuss the relation to A G,S , X G,S , P G,S constructions.In Section 5, we close with a number of examples illustrating formulas for charts and flips.
Acknowledgements.
We are grateful to David Ben-Zvi for furnishing us our lodestar, andto Sam Gunningham for countless helpful discussions and clarifications around the categoricalapproach to algebraic geometry in which we obtained our results.The work of D.J. and A.S. was supported by European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (grant agreementno. 637618). D.J. was partially supported by NSF grant DMS-0932078, administered by theMathematical Sciences Research Institute while in residence at MSRI during the QuantumSymmetries program in Spring 2020. A.S. was partially supported by the NSF PostdoctoralFellowship DMS-1703183, as well as by the International Laboratory of Cluster GeometryNRU HSE grant no. 2020-220-08-7077.2.
Decorated surfaces and stratified factorization homology
The basic observation motivating the interaction between character stacks and factorizationhomology is as follows: the assignment to each decorated surface of its decorated characterstack is functorial for stratified embeddings and their isotopies, and characterized by a strong
G T GTB
Figure 5.
Disks with parabolic structures D G , D T , D B .locality property called excision; meanwhile factorization homology provides universal suchfunctorial invariants. These observations are captured in the following definitions. Definition 2.1.
Let S and S be decorated surfaces. A stratified embedding ι : S ã Ñ S isan oriented embedding of surfaces which preserving the partition into walls, regions and labels.A stratified isotopy is a continuous path through the space of stratified embeddings. Definition 2.2.
The symmetric monoidal p , q -category S urf is a category with Objects: decorated surfaces, stratified embeddings of decorated surfaces, stratified isotopies of stratified embeddings.The symmetric monoidal structure is given by disjoint union.
Definition 2.3.
The full symmetric monoidal (2,1)-subcategory D isk Ă S urf of basic deco-rated disks is generated by the following decorated surfaces shown on Fig. 5: D G = R with no walls, the unique region labeled G ; D T = R with no walls, the unique region labeled T ; D B = R with the x -axis as its unique wall, the upper region labeled G , and the lowerregion labeled T . Remark 2.4.
The disks above are a basis for the topology of decorated surfaces, in the sensethat any point of a decorated surfaces sits in a neighborhood of the form D G , D T or D B . Definition 2.5. A D isk-algebra is a symmetric monoidal functor F : D isk Ñ Pr .Here, Pr is the symmetric monoidal (2,1)-category whose objects are presentable k -linearcategories, morphisms are colimit-preserving functors, and 2-morphisms are natural isomor-phisms. Notation 2.6.
Let C “ p C , b , σ q be a braided monoidal category with tensor multiplicationand braiding denoted by b : p V, W q ÞÝÑ V b W and σ V,W : V b W ÞÝÑ W b V respectively. From C we obtain a pair of braided monoidal categories C op “ p C , b op , σ op q and C “ p C , b , σ q , where b op is the opposite tensor product: b op : p V, W q ÞÝÑ W b V, while σ op V,W “ σ ´ V,W and σ V,W “ σ ´ W,V . In fact these are equivalent as braided tensor categories, via the identity functor with tensorstructure given by the braiding.
UANTUM DECORATED CHARACTER STACKS 19
Definition 2.7.
Given a pair A , B of braided tensor categories, an p A , B q -central structureon a tensor category C is a braided tensor functor A b B Ñ Z Dr p C q .Note that by definition of the Drinfeld center, the data of a lift a monoidal functor F : A b B Ñ C to an p A , B q -central structure is equivalent to the data, for every object of A b B of a half-braiding , i.e., a natural isomorphism F p a b b q b c – c b F p a b b q , whichmust be natural in all arguments, and moreover compatible with the braiding on A b B inthe obvious way. Proposition 2.8. A D isk -algebra is uniquely determined by the following data:(1) The categories F p D G q , F p D T q , F p D B q P Pr ,(2) The structures of a ribbon braided tensor category on F p D G q and F p D T q ,(3) The structure of a p F p D G q , F p D T qq -central tensor category on F p D B q . We refer to Section 2.2 for a definition of the notion of a central tensor category, and forexamples.
Definition 2.9 (Adapted from [AFT17]) . The stratified factorization homology with coeffi-cients in F is the functor, Z : S urf Ñ Pr defined as the unique left Kan extension of the functor F from the diagram D isk (cid:34) (cid:34) F (cid:47) (cid:47) Pr S urf Z (cid:60) (cid:60) In other words, Z p S q is defined as the following homotopy colimit: Z p S q “ colim X Ñ S Z p X q , taken in the 2-category of locally presentable categories, indexed over all embeddings and theirisotopies, of disjoint unions X of finitely many copies of D G , D B , and D T , into Σ. Unpackingthe definition of colimit, this means:(1) For each disjoint union of decorated disks, X “ D \ IG \ D \ JB \ D \ KT , we have Z p X q “ F p D G q I b F p D B q J b F p D T q K . (2) Each stratified embedding i X : X Ñ S , induces a colimit-preserving functor, i X ˚ : F p D G q I b F p D B q J b F p D T q K Ñ Z p S q , (3) Each stratified isotopy induces an isomorphism of induced functors, and(4) Each factorization i X : X Ñ X Ñ S of a disk embedding through an intermediatedisk embedding (i.e. where X , X P D isk), induces a factorization of the functor i X ˚ through the fixed braided tensor, tensor, and central structures on F p D G q , F p D B q ,and F p D T q . Likewise, and isotopies thereof are given by the fixed braiding, associator,and action morphisms. Strictly speaking, the labeling of regions of a decorated surface is additional data on the stratification (whileit is also less general because we only allow simple smooth curves as defects). However, the labelling data doesnot materially change any argument from [AFT17], so we will simply adapt this definition and the formulationof the excision theorem to our setting, without further comment. (5) Finally, the category Z p S q is universal for such structures: given any other category C equipped with all of the above data, we obtain a canonical functor Z p S q Ñ C , suchthat the data on C is obtained from that on Z p S q by composition.2.1. Cylinder algebras, module actions and excision.
The definition of factorizationhomology as a left Kan extension emphasizes universal properties and topological invariance.These are useful in applications, but are not convenient for doing explicit computations. Letus now recall the excision formula from [AFT17], which gives a concrete algebraic algorithmfor doing computations.
Definition 2.10.
A decorated 1-manifold X is an oriented 1-manifold with a finite collection P of points. The walls and regions of X are P and the connected components of X (cid:114) P ,respectively. We fix the further data of a labeling of the regions from the alphabet t G, T u ,such that two regions sharing a wall have distinct labelings. Definition 2.11.
A decorated 1-disk is a disjoint union of the four decorated intervals inFig. 6.
G T G TB T GB
Figure 6.
Flagged 1-manifolds are disjoint unions of the above.
Definition 2.12. An X -gate in a decorated surface is a neighborhood U of the boundary witha homeomorphism U – X ˆ r , q with a cylindrical decorated half-surface. We abbreviatethe cases U “ D G , D T , D B , by G -gate, T -gate, B -gate respectively.Concatenation in the p , q direction makes any cylindrical decorated surface X ˆ p , q intoan algebra object in S urf, and hence equips the category Z p X ˆ p , qq with the structure of amonoidal category. Similarly, the specification of an X -gate on a decorated surface S induceson Z p S q the structure of a X ˆ p , q -module category.A decomposition of the decorated surface S , denoted S “ S \ X ˆp , q S , consists of the choiceof a cylindrical decorated surface embedding X ˆ p , q ã Ñ S , such that S (cid:114) p X ˆ p , qq is adisjoint union of decorated surfaces S and S . In this case, S and S are naturally markedwith X -gates. Theorem 2.13 (Excision, [AFT17]) . Each decomposition of a decorated surface, S “ S \ X ˆp , q S , induces a canonical equivalence of categories, Z p S q » Z p S q b Z p X ˆp , qq Z p S q . Recollection about quantum groups.
In this section we recall the braided tensorcategories Rep q p G q and Rep q p T q , and the tensor category Rep q p B q , consisting of ‘integrable’representations of the quantized universal enveloping algebras U q p g q , U q p t q and U q p b q , and usethem to define quantizations of framed configuration spaces. We will not recall in detail explicitpresentations of each enveloping algebra in this paper, referring to standard texts. However,we recall more details the cases SL and PGL , in order to fix notation for computations tocome. UANTUM DECORATED CHARACTER STACKS 21
Given a lattice Λ, we let Vect Λ denote the category of Λ-graded vector spaces, whose objectsare vector spaces V equipped with a decomposition V “ ‘ λ P Λ V λ , and whose morphisms arethose linear maps respecting the Λ-grading. The simple objects of this category are of theform C λ (a one-dimensional vector space supported at λ ), and every object is a direct sum ofsimples. Clearly, we have isomorphism C λ b C µ Ñ C λ ` µ .Given, further, a complex number q P C and an integer-valued symmetric bilinear form, α : Λ ˆ Λ Ñ Z , we obtain a braided tensor category structure on Vect Λ by declaring the braiding on C λ b C µ to be scalar multiplication q α p λ,µ q followed by the flip of tensor factors, and extending toarbitrary direct sums of simples.Fix a reductive algebraic group G along with a Borel subgroup B Ă G , and denote by T the universal Cartan quotient. Definition 2.14.
The category Rep q p T q denotes the braided tensor category p Vect Λ , ´ α q ,where Λ is the weight lattice and α is the Cartan pairing. Notation 2.15.
Let U q p g q denote the quantized universal enveloping algebra of g “ Lie p G q ,defined over the field C p q q . Denote the standard Serre generators by E , . . . , E r , F , . . . F r ,and K λ , for λ P Λ. Let U q p b q (resp., U q p b ` q ) denote subalgebras generated K λ ’s and by F i ’s(resp. E i ’s). Let U q p t q denote the subalgebra generated by the K λ ’s.Let Λ ad Ă Λ denote the root lattice. In order to determine G from g , we need to furtherspecify a lattice Λ G between them, Λ ad Ă Λ G Ă Λ. We then define Rep q p T q as the fullsubcategory of the category of U q p t q -modules: its representations are spanned by weightvectors v µ , µ P Λ, on which K λ acts by q α p λ,µ q . Recall that a representation of an algebra A is called locally finite if the orbit A ¨ v of every vector v P V is finite-dimensional. Definition 2.16.
The category Rep q p B q is the full subcategory of locally finite U q p b q -moduleswhose restriction to U q p t q lies in Rep q p T q .Recall that for generic q , the category Rep q p G q is in fact semisimple: every object may bepresented as a (possibly infinite) direct sum of simple finite-dimensional U q p g q -modules. Itfollows from the classification of finite-dimensional U q p g q -modules that the restriction of anobject of Rep q p G q to U q p b q lies in Rep q p B q .The universal R -matrix is an invertible element R lying in a completion of U q p g q b U q p g q ,satisfying the quantum Yang-Baxter equation. In Sweedler notation, we write R “ R p q b R p q .Then the braiding is given by: σ G p v b w q “ ` R p q w ˘ b ` R p q v ˘ Definition 2.17.
The category Rep q p G q is the category of locally finite U q p g q -modules, all ofwhose weights lie in Λ G , with its braided tensor structure given via the universal R -matrix.Recall that in Dynkin type A m we have Λ “ Z , and Λ ad “ Z , hence the only twobraided tensor categories we consider are Rep q p SL q (with Λ SL “ Z ), and Rep q p PGL q (withΛ PGL “ Z ). Let us recall these in detail now.The quantum group U q p sl q is a quasi-triangular Hopf algebra over C p q q with generators (cid:32) E, F, K ˘ ( subject to relations KE “ q EK, KF “ q ´ F K, r E, F s “ K ´ K ´ q ´ q ´ . It carries the comultiplication∆ p E q “ E b ` K b E, ∆ p F q “ F b K ´ ` b F, ∆ p K q “ K b K, (2.1)the antipode S p E q “ ´ K ´ E, S p F q “ ´ F K, S p K q “ K ´ , and the counit (cid:15) p K q “ , (cid:15) p E q “ (cid:15) p F q “ . The universal R -matrix of the quantum group can be written as R G “ Ψ q ` ´p q ´ q ´ q E b F ˘ q ´ H b H , (2.2)where Ψ q p x q is the quantum dilogarithmΨ q p x q “ ź a “ ` q a ´ x . The latter is a close relative with the q-exponential, namelyΨ q p x q “ exp q ˆ xq ´ q ´ ˙ where exp q p x q “ ÿ n “ x n r n s q ! with r n s q ! “ n ź a “ q a ´ q ´ . The category Rep q p SL q and its subcategory Rep q p P GL q are ribbon categories, with theribbon element ν G acting in the n –dimensional irreducible representation V n of U q p sl q by thescalar q n ` .The quantum group U q p sl q has Borel subalgebras U q p b ˘ q generated by (cid:32) E, K ˘ ( and (cid:32) F, K ˘ ( respectively, their nilpotent subalgebras U q p n ˘ q generated respectively by theelements E and F , and the torus U q p t q “ U q p b ` q X U q p b ´ q generated by the elements K ˘ .Note that R -matrix that determines the braiding on Rep q p T q from Definition 2.14 may beexpressed as R T “ q ´ H b H . The corresponding ribbon element ν T acts on the 1-dimensional simple C n , n P Z by the scalar q n .Let us describe the different categories which arise in these cases completely explicitly, forgeneric q . The category Rep q p SL q consists of arbitrary direct sums of simple representations V p k q with highest weight k , Rep q p B q consists of direct sums of indecomposable modules X p k, l q , with k ” l mod 2, where X p k, l q has a basis t v k , v k ` , . . . v l u , each v i a weight vectorof weight i , with F v i “ v i ´ . In this case Rep q p T q is the category of Z -graded vector spaceswith the pairing x m, n y “ ´ mn . All three corresponding categories are the same for PGL inplace of SL , except we add a requirement that all T -weights (in particular k and l above)are even, i.e. in Λ ad . UANTUM DECORATED CHARACTER STACKS 23
The D isk -algebra Rep q p T Ð B Ñ G q and Z p S q . The inclusion i : U q p b q Ñ U q p g q andthe projection π : U q p b q Ñ U q p t q (obtained by quotienting by F i “ q induce pullback functors, i ˚ : Rep q p G q Ñ Rep q p B q , π ˚ : Rep q p T q Ñ Rep q p B q . These functors admit the important additional structure of a central lift (see Section 2.2),making Rep q p B q into a p Rep q p G q , Rep q p T qq -central tensor category . Indeed, since the R -matrix (2.2) is a sum of elements of U q p b ` q b U q p b q , it has a well-defined action on anytensor product i ˚ V b W for objects V, W of Rep q p G q and Rep q p B q respectively, and composingthis action with the flip of tensor factors yields a morphism in Rep q p B q . Moreover, givenweight vectors v λ P π ˚ χ and w µ P W , the form of the coproduct Eq. (2.1) along with the factthat π p F i q “ F i ¨ p v λ b w µ q “ v λ b F i w µ , where F i w µ has weight µ ´ α i . Hence σ T ˝ F i p v λ b w µ q “ q p λ,µ ´ α i q F i w µ b v λ “ q p λ,µ q F i w µ b K ´ i v λ “ F i ¨ σ T p v λ b w µ q , and we see that σ T also defines a morphism in Rep q p B q . Definition 2.18.
We define a central structure, i ˚ b π ˚ : Rep q p G q b Rep q p T q ÝÑ Z p Rep q p B qq V b χ ÞÝÑ i ˚ p V q b π ˚ p χ q via the half-braiding, σ p V,χ q ,W : p i ˚ p V q b π ˚ p χ qq b W σ p q ˝ R G ˝p R T q ´ ÝÝÝÝÝÝÝÝÝÝÝÝÑ W b p i ˚ p V q b π ˚ p χ qq . Definition 2.19.
The parabolic induction D isk -algebra Rep q p T Ð B Ñ G q is determinedvia Proposition 2.8 by the assignments: D G ÞÑ Rep q p G q , D T ÞÑ Rep q p T q , D B ÞÑ Rep q p B q , with their braided tensor and central structures defined above.We now come to the main definition of the paper, which as been stated already in theintroduction. Definition 2.20.
The quantum decorated character stack Z p S q is the stratified factor-ization homology, Z p S q : “ ż S Rep q p T Ð B Ñ G q , of S with coefficients in the parabolic induction algebra Rep q p T Ð B Ñ G q . Monadic reconstruction of module categories
Monadic reconstruction for module categories is a center-piece in the higher representationtheory of tensor categories. It was developed the furthest in the fusion, and in the finitenon-semisimple, settings; a comprehensive reference is [EGNO16]. Monadic reconstructionwas developed also in the locally presentable framework in [BBJ18a; BBJ18b; BJS18]. Werecall those constructions here, with an additional wrinkle: in contrast to the usual situation,generators we encounter will not be projective, even in the relative sense; this reflects that theframed moduli spaces at the classical level are only quasi-affine rather than affine. For thisreason, in place of the equivalences of categories featuring in usual monadic reconstructionwe will instead obtain open embeddings of categories, and we will compute their orthogonalcomplements.3.1.
Adjoints and monads.
To begin, we recall some standard facts and definitions inenriched category theory.
Definition 3.1 (Sketch) . A k -linear category C is locally presentable if the colimit colim p D q of any diagram D in C exists as an object of C , and if C is generated under κ -filtered colimitsby a small category of κ -compact objects, for some cardinal κ . Remark 3.2.
It will suffice to take κ “ ℵ , the countable cardinal; such categories are called locally finitely presentable . Definition 3.3.
Let Pr denote the (2,1)-category of locally presentable k -linear categorieswith colimit-preserving functors and natural isomorphisms. Remark 3.4.
Examples of locally finitely presentable categories include the category of A -modules for an arbitrary associative k -algebra, the category of representations of an algebraicgroup, and the category of quasi-coherent sheaves on algebraic variety, or more generally onan Artin stack. All examples we will consider will turn out to one of these types of categories,or its q -deformation. Theorem 3.5 (Special adjoint functor theorem) . Any colimit-preserving functor L : C Ñ D between locally finitely presentable categories has a right adjoint R , however R is not itselfnecessarily colimit-preserving. We will refer below to the adjoint pair p L, R q , and to the unit η : id C Ñ LR and counit (cid:15) : RL Ñ id D of the adjunction. Definition 3.6.
The functor G : D Ñ C is conservative if it reflects isomorphisms, i.e. forany morphism f in D , G p f q is an isomorphism if, and only if, f is an isomorphism. Proposition 3.7.
Suppose that C and D are abelian locally presentable categories and G : D Ñ C is colimit-preserving. Then G is conservative if, and only if, G p X q “ im-plies X “ for any object X . Definition 3.8.
The functor L : C Ñ D is dominant if D is generated under colimits by theimage of L . Proposition 3.9.
The functor L : C Ñ D is dominant if, and only if, its right adjoint R : D Ñ C is conservative. Definition 3.10.
Let L : C Ñ D be a colimit preserving functor, and let R denote its rightadjoint. The monad for the adjunction is the composite functor A “ RL : C Ñ C , equipped UANTUM DECORATED CHARACTER STACKS 25 with the structure of a unital algebra in End p C q , with unit given by the unit η : Id C Ñ RL defining the adjunction, and with multiplication, AA “ RLRL id R ˝ (cid:15) ˝ id L ÝÝÝÝÝÝÑ RL “ A, obtained from the counit of the adjunction. Definition 3.11. An A -module in C is a pair p c, ψ q of an object c P C and a morphism ψ : Ac Ñ c , satisfying an evident associativity axiom which we omit. The Eilenberg-Moore category A ´ mod C is the category of A -modules in C , with the evident notion of morphisms. Definition 3.12.
The comparison functor r R : D Ñ A ´ mod C sends d P D to the object R p d q , equipped with an A -module structure via AR p d q “ RLR p d q id R ˝ (cid:15) ÝÝÝÑ d. Theorem 3.13 (Barr-Beck reconstruction) . If the functor R is conservative and colimitpreserving, then r R defines an equivalence of categories. D » A ´ mod C . We will require a variant of Barr-Beck reconstruction in the abelian category setting,involving a weaker condition and a weaker conclusion. We were unable to find a reference forthe claim, although it uses standard constructions in the theory of abelian categories.
Definition 3.14.
We say that a fully faithful embedding i : D Ñ E is reflective (resp. open ),and that D is a reflective subcategory (resp. open subcategory ), if i admits a left adjoint(resp. exact left adjoint). Lemma 3.15 ([BW05] p. 111, Corollary 7 and Theorem 9) . The comparison functor r R isfaithful if, and only if, the counit of p L, R q is an epimorphism, and is fully faithful if, andonly if, the counit of the adjunction is a regular epimorphism. Theorem 3.16.
Suppose that C and D are abelian categories, and that R is conservative (butnot necessarily colimit-preserving). Then r R defines a reflective embedding of D into A ´ mod C . Proof.
By the previous discussion, we need to show that the counit of p L, R q is a regular epi-morphism and that r R admits a left adjoint. Let us fix the forgetful functor For : A ´ mod C Ñ C .The functor R factors as For ˝ r R . Since R is limit-preserving and conservative, it is faithful,hence so is r R . By Lemma 3.15, the counit for the adjunction is an epimorphism. In an abeliancategory every epimorphism is regular, hence again by Lemma 3.15, r R is in fact fully faithful.We also have that r R is limit preserving–because R is limit-preserving and limits in A ´ mod C are computed in C –hence r R admits a left adjoint r L . (cid:3) In summary, we have the following commutative diagram, in which each inner arrow is theright adjoint to each outer arrow, and in which the inner triangle and outer triangle each arecommuting triangles.
D C A ´ mod C RL r R r L ForFree
Remark 3.17.
A formula for the left adjoint r L may be given as follows. Any M P A ´ mod C may be presented as coker p Free p V q f ÝÑ Free p W qq , for objects V , W of C . Since L is colimitpreserving, we have: r L p M q “ coker ˆ p r L ˝ Free qp V q r L p f q ÝÑ p r L ˝ Free qp W q ˙ “ coker ˆ L p V q r f ÝÑ L p W q ˙ , (3.1)where r f is the determined by f by the following natural maps,Hom A ´ mod C p RL p V q , RL p W qq „ ÝÑ Hom C p V, RL p W qq L ÝÑ (3.2) L ÝÑ Hom D p L p V q , LRL p W qq (cid:15) ÝÑ Hom D p L p V q , L p W qq Definition 3.18.
Let C be an abelian category, and let D be a full subcategory. The leftorthgonal K D is the full subcategory of C consisting of objects c such that Hom p c, d q “
0, forall d P D . Similarly, the right orthogonal D K is the full subcategory of C consisting of objects c such that Hom p d, c q “
0, for all d P D . Proposition 3.19.
Suppose that R : D Ñ E defines a reflective embedding, with left adjoint L . Then we have an identification, R p D q “ p ker p L qq K , between the essential image of R and the right orthogonal to the kernel of L . Proof.
Let X P E . By the Yoneda embedding, L p X q “ p L p X q , Y q “ Y P D , which is if, and only if, Hom p X, R p Y qq “ Y P D . The first conditionis that X P ker p L q , while the second condition is that X P K R p D q . Hence ker L “ K R p D q ,hence p ker L q K “ p K R p D qq K “ R p D q . (cid:3) Internal homomorphisms.
The preceding discussion of monadicity is especially usefulin analyzing rigid tensor categories and their module categories. We recall these applicationshere.Let A be a rigid tensor category, and M its module category, and fix an object m P M .Then the functor act m : A Ñ M sending a ÞÑ a b m is colimit preserving. Definition 3.20.
The internal homomorphisms from m to n is the object,Hom A p m, n q “ act Rm p n q P A . In what follows, we will often write Hom p m, n q instead of Hom A p m, n q , if there is no ambiguityin regards to the tensor category. Definition 3.21.
The evaluation morphism ev m,n P Hom p Hom p m, n q b m, n q is the image of the identity morphism, id Hom p m,n q , under the canonical isomorphismHom p Hom p m, n q b m, n q » End p Hom p m, n qq . Definition 3.22.
The composition law
Hom p n, o q b Hom p m, n q ÝÑ Hom p m, o q is the image of the morphismev n,o ˝ p id Hom p n,o q b ev m,n q P Hom p Hom p n, o q b Hom p m, n q b m, o q UANTUM DECORATED CHARACTER STACKS 27 under the canonical isomorphismHom p Hom p n, o q b Hom p m, n q b m, o q » Hom p Hom p n, o q b Hom p m, n q , Hom p m, o qq Definition 3.23.
The internal endomorphism algebra of m isEnd p m q “ Hom p m, m q P A , with the algebra structure given by the composition law. Remark 3.24.
Let A denote the monad associated to the adjoint pair p act m , Hom p m, ´qq in the preceding section. Then we have a natural isomorphism End p m q – A p A q , and thealgebra structure on End p m q is the one induced from the monadic structure on A . Hencethe calculus of internal Hom’s is a special case of monadic reconstruction, so we are able toproduce genuine algebra objects in A , rather than in End p A q . This is due to the fact thatHom p m, ´q and act m , and hence their composition are A -module functors, and we have theusual equivalence End A p A q “ A op . Definition 3.25.
We say that m is an A -generator of M if Hom p m, ´q is conservative, andwe say that m is A -compact-projective if Hom p m, ´q is colimit preserving.We have the following translation of Theorem 3.13 and Theorem 3.16 to the case of modulecategories. Theorem 3.26 (Barr-Beck reconstruction for module categories [Ost03],[BBJ18a]) . Fix A arigid tensor category, M its abelian module category, m an object of M .(1) Suppose that m is an A -generator of M . Then we have a reflective embedding, M ã Ñ End p m q´ mod A , realizing M as the reflective subcategory of End p m q´ mod A .(2) Suppose, moreover, that m is an A -compact-projective generator. Then we have anequivalence, M » End p m q ´ mod A . Remark 3.27.
In all examples we will consider in the paper, the required left adjoint isclearly exact (e.g. if A “ Rep q p G q or Rep q p T q , so that we obtain an open embedding. Proposition 3.28.
Suppose C and D are rigid tensor categories, F : C Ñ D is a tensorfunctor, and suppose that d P D is an C -projective generator, so that D » End p d q´ mod C .Suppose further that M is a D -module category, with a D -projective generator, m . Then wehave D » End p d q´ mod C , M » End p m q´ mod D , and we obtain a homomorphism of algebras in C : End C p d q Ñ End C p m q , such that the action of D on M is given by relative tensor products over End C p d q . Monadic reconstruction: classical examples.
Let us recall some very classical andwell-known facts in geometric representation theory of flag varieties, recast in the categoricalframework of noncommutative algebraic geometry and treated by monadic reconstruction.
Example 3.29.
Let us start with a simple, almost trivial, application of the Barr-Becktheorem. Let R Ñ S be commutative algebras. Let X “ Spec p R q , Y “ Spec p S q so thatwe have the corresponding map of varieties π : Y Ñ X . We have QC p X q “ R ´ mod and QC p Y q “ S ´ mod. There is a natural functor L : “ π ˚ : QC p X q Ñ QC p Y q given by M Ñ M b R S . The rightadjoint is R : “ π ˚ : QC p Y q Ñ QC p X q . Note that QC p Y q is a module category for QC p X q .Barr-Beck reconstruction realizes QC p Y q as the category of objects of QC p X q equipped withan action of the monad RL .The monad for the adjunction is ´ b R S . An algebra for this monad is exactly an R -modulewhich has a compatible S -module structure. Clearly such an object in QC p X q corresponds toan object in QC p Y q .The examples we consider throughout this paper build on these examples in three distinctdirections.(1) X and Y will be replaced by stacks. Hence we will work internally to a monoidalcategory which carries the symmetries of the stack.(2) The map Y Ñ X will not be affine, but quasi-affine. Hence applying Barr-Beck willonly give open embeddings of categories rather than equivalences.(3) We will be interested in quantizations; hence the symmetric monoidal categories ex-pressing classical symmetries will be replaced systematically by their braided monoidal q -deformations.In the remainder of this section, we will illustrate the first two points with classical examples.In the next section, we treat the quantum case. Example 3.30.
Let X “ ‚{p G ˆ G q and let Y “ ‚{ G be the stack quotients of the one-pointvariety, by the trivial group action – these are sometimes called classifying stacks , anddenoted B p G ˆ G q , BG . By definition, we have QC p X q “ Rep p G q b Rep p G q , QC p Y q “ Rep p G q . The group homomorphism g Ñ p g, g ´ q induces a natural map π : Y Ñ X . Pullback along π ˚ defines a Rep p G q b Rep p G q -module structure on Rep p G q , and we can apply Barr-Becktechniques to describe it. Let G be the trivial G -module. The functor L “ π ˚ is given byact G , or p V b W q ÞÝÑ V b W. The right adjoint can be computed, R “ π ˚ V ÞÝÑ à λ P P ` p V b V ˚ λ q b V λ . One then computes an isomorphism of algebras,End G ˆ G p G q “ à λ P P ` V ˚ λ b V λ – O p G q . Hence, Barr-Beck asserts that Rep p G q is equivalent to the category O p G q´ mod G ˆ G . Inother words, O p G q is an algebra object in Rep p G q b Rep p G q , and Rep p G q can be realized asRep p G q b Rep p G q -modules which have the structure of an algebra over the Rep p G q b Rep p G q -algebra O p G q . The equivalence in this case expresses the obvious isomorphism of stacks, G {p G ˆ G q » ‚{ G .Note that the map Y Ñ X is a fibration with fiber G . Barr-Beck for the map of varieties G Ñ ‚ is easy–this is just a map of affine varieties as in the previous example. Thus we arereally just revisiting Example 3.29 in a family over the stack X “ ‚{p G ˆ G q . Example 3.31.
Let Y “ N z G and let X be a point. The left adjoint L “ π ˚ takes avector space V to the trivial vector bundle over Y with fiber V . The right adjoint R “ π ˚ is the functor of global sections. Barr-Beck does not allow us to reconstruct QC p Y q inside UANTUM DECORATED CHARACTER STACKS 29 QC p X q , because QC p Y q does not have a compact-projective generator. The structure sheaf isa generator, but the global sections functor Hom p O , ´q does not preserve colimits (it is notright exact). Therefore, we have an open embedding QC p N z G q ã Ñ O p N z G q´ mod . Geometrically, N z G is quasi-affine, so it sits as an open set in N z G “ Spec p O p N z G qq , andthe orthogonal complement to QC p N z G q in QC p N z G q is the Serre subcategory of torsionsheaves supported on the complementary closed subvariety. Example 3.32 (Classical parabolic induction) . Let X “ ‚{p G ˆ T q and let Y “ ‚{ B . Themap B Ñ G ˆ T induces a natural map π : Y Ñ X . Recall that π ˚ equips QC p Y q “ Rep p B q with the structure of a QC p X q “ Rep p G ˆ T q -module category. Let B denote the trivialrepresentation of B . It is a Rep p G ˆ T q -generator, but not Rep p G ˆ T q -compact projective.Then we have the module functor, L “ act B : Rep p G ˆ T q ÝÑ Rep p B q , V b χ ÞÝÑ i ˚ V b π ˚ χ. We then have End G ˆ T p B q » à λ P P ` V λ b C ´ λ » O p N z G q , with the latter isomorphism being one of algebras in Rep p G ˆ T q . Thus Rep p B q can berealized as an open subcategory of G ˆ T -equivariant O p N z G q -modules. By the classicalProj-construction applied to the T -graded ring O p N z G q , this open subcategory is furtherequivalent to the category of G -equivariant sheaves on the flag variety B z G . Thus the monadiccomparison functor is nothing but the classical parabolic induction equivalenceRep p B q » QC G p B z G q , M ÞÝÑ M ˆ B G, sending a B -module M to the G –equivariant vector bundle M ˆ B G on B z G . Note that theleft adjoint to this functor is given by taking the fiber F e of a G –equivariant sheaf F on B z G over the identity element e P G ; this fiber carries a B -action since the identity coset Be is afixed point for the action of B on B z G by right translation.3.4. Monadic reconstruction:
Rep q p G q and F q p G q . As we shall now explain, the monadicapproach to the classical parabolic induction construction carries over equally well to thequantum setting.
Notation 3.33.
In what follows we setHom H p V, W q : “ Hom
Rep q p H q p V, W q for H being G , B , or T . We do the same for Rep q p H q op and Rep q p H q , so that e.g.End G b G p G q “ End
Rep q p G q b Rep q p G q p G q . Example 3.34.
Consider the rigid tensor category A “ Rep q p G q op b Rep q p G q . Then M “ Rep q p G q is a module category for A with the module structure given by A b M ÝÑ M , p V b W q b U ÞÝÑ V b U b W. Writing G P M for the unit U q p g q -module, we obtain the functoract G : A ÝÑ M , V b W ÞÝÑ V b W. Note that Hom G op ˆ G ` V λ b V µ , End G op ˆ G p G q ˘ “ Hom G op ˆ G ` V λ b V µ , act R G p G q ˘ » Hom G p V λ b V µ , G q . Since the category Rep q p G q is semisimple, andHom G p V λ b V µ , G q “ C if V λ » V ˚ µ , , it follows that End G op ˆ G p G q » à V P Irr p M q V ˚ b V “ à λ P Λ ` V ˚ λ b V λ , where the first direct sum is taken over the isomorphism classes of simple objects in M . Thealgebra structure on End G ˆ G op p G q is given by p ξ b v q b p ζ b w q “ p ξ b op ζ q b p v b w q “ p ζ b ξ q b p v b w q . It follows that End G op ˆ G p G q is isomorphic as an algebra in A to the Faddeev–Reshetikhin–Takhtadjan quantum coordinate ring F q p G q , the Hopf algebra of matrix coefficients of finite-dimensional type I representations of U q p g q . The algebra F q p G q is a q –deformation of thePoisson algebra of functions on the simple Lie group G equipped with its standard Poisson-Liestructure π “ r L ´ r R given by the difference of the left and right translates of the classical r –matrix, see [Sem92]. Remark 3.35.
Let l λ Ă V λ and l ´ λ Ă V ˚ λ be the highest and lowest weight lines respectively,and consider the principal minor f λ “ v ˚´ λ b v λ P F q p G q , where v ˚´ λ , v λ are any elementsof l ´ λ , l λ normalized such that @ v ˚´ λ , v λ D “
1. Then since V λ ` µ appears with multiplicity onein V λ b V µ and v λ b v µ spans the corresponding highest weight line, we have f λ f µ “ f λ ` µ . Inparticular, the subalgebra of F q p G q generated by the principal minors t f λ u λ P Λ ` is commutative,and isomorphic to the polynomial ring C p q qr Λ ` s . Example 3.36.
Let us now consider the rigid tensor category A “ Rep q p G q b Rep q p G q . Then M “ Rep q p G q again has the structure of an A –module category, with the module structurenow being given by A b M ÝÑ A , p V b W q b U ÞÝÑ V b W b U. The key difference in this case is the nontrivial structure map coming from the braiding inRep q p G q : we have the isomorphism σ W ,V : V b W b V b W b U ÝÑ V b V b W b W b U. (3.3)The same argument used in the previous example shows that as an object of A , we again haveEnd G p G q » à λ P Λ ` V ˚ λ b V λ , but in view of (3.3) the algebra structure is now given by p ξ b v q b p ζ b w q “ σ V ˚ ,W ˚ p ξ b ζ q b p v b w q“ p R ¨ ζ b R ¨ ξ q b p v b w q . The algebra End G p G q thus coincides with F q p G ` q , the q –deformation of the Poisson algebraof functions on the simple Lie group G equipped with the Poisson structure π ` “ r L ` r R givenby the average of the left and right translates of the classical r –matrix, see again [Sem92]. UANTUM DECORATED CHARACTER STACKS 31
Remark 3.37.
Both algebras F q p G q and F q p G ` q have subalgebras F q p N z G q , F q p N z G ` q given by à λ P P ` l ´ λ b V λ Ă à λ P P ` V ˚ λ b V λ , where l ´ λ is the lowest weight line in V ˚ λ . However, while multiplication rule in F q p N z G q issimply given by p ξ ´ λ b v q b p ζ ´ µ b w q “ p ζ ´ µ b ξ ´ λ q b p v b w q , in F q p N z G ` q we have p ξ ´ λ b v q b p ζ ´ µ b w q “ q ´p λ,µ q p ζ ´ µ b ξ ´ λ q b p v b w q , owing to the fact that the R –matrix acts on the tensor product of lowest-weight vectors by R G p ξ ´ λ b ζ ´ µ q “ q ´p λ,µ q ξ ´ λ b ζ ´ µ . Nonetheless, the half–ribbon element ν T from Rep q p T q induces an isomorphism of algebras in Rep q p T ˆ G q ν T b id : F q p N z G ` q Ñ F q p N z G q , ξ ´ λ b v ÞÝÑ q p λ,λ q ξ ´ λ b v. Example 3.38.
The Rep q p G q –module category in the Example 3.36 can be understood asthe pullback of a Rep q p G q –module structure by the tensor functor m G : Rep q p G q b Ñ Rep q G ,with the tensor structure being given by the braiding in Rep q p G q . Such pullbacks will ariseoften for us as we open additional gates within G - and T -regions. Of particular importance isthe case of a Rep q p T q –module category of the form M “ A ´ mod T , regarded as a Rep q p T q –module category via m T with tensor structure coming from the braiding in Rep q p T q . ByProposition 3.28, we have an equivalence M » r A ´ mod T , where r A “ End T p A q . As an object of Rep q p T q , r A may be determined by a similar argument to that made inExample 3.34. Writing A “ À λ P Λ A λ b C λ , we find thatHom T ´ C λ b C µ , r A ¯ » Hom A ´ mod T p C λ ` µ b A, A q » A λ ` µ . It follows that we have r A “ à λ,µ P Λ A λ ` µ b C λ b C µ , (3.4)and recalling (3.3) we see that the algebra structure reads r A λ,µ b r A ν,ρ ÝÑ r A λ ` ν,µ ` ρ p a b χ λ b χ µ q ¨ p a b χ ν b χ ρ q “ p a a q b ´ m b T ˝ σ ´ C ν b C µ ¯ p χ λ b χ µ b χ ν b χ ρ q“ q ´p µ,ν q a a b p χ λ ` ν b χ µ ` ρ q . (3.5)3.5. Monadic reconstruction:
Rep q p B q and F q p N z G q . We are now ready to give amonadic description of the category Rep q p B q in terms of the quantized flag variety, byanalogy with the classical parabolic induction equivalence described in Example 3.32. Thefirst step in this direction is the computation outlined in the following example. Example 3.39.
Recall that Rep q p B q is a module category for Rep q p G ˆ T q . We write B for the monoidal unit in Rep q p B q , and consider the functoract B : Rep q p G ˆ T q ÝÑ Rep q p B q , V b χ ÞÝÑ i ˚ V b π ˚ χ. Its right adjoint is given byact R B : Rep q p B q ÝÑ Rep q p G ˆ T q ,M ÞÝÑ à λ P Λ ` µ P Λ Hom B p V λ b C µ , M q b V λ b C µ , which implies act R B p M q » p M b F q p G ˆ T qq U q p b q , where we have set F q p G ˆ T q “ à λ P Λ ` µ P Λ ` C ˚ µ b V ˚ λ ˘ b V λ b C µ , with U q p b q acting diagonally in the C ´ µ b V ˚ λ tensor factor, and U q p g q b U q p t q acting in V λ b C µ . Let us now turn our attention to the braided-commutative algebra End G ˆ T p B q . Setting M “ B in the formula above, we obtain an isomorphism in Rep q p G ˆ T q End G ˆ T p B q » F q p G ˆ T q U q p b q “ à λ P P ` p C ˚´ λ b l ´ λ q b V λ b C ´ λ where l ´ λ is the lowest weight line in V ˚ λ . On the other hand, recall from Remark 3.37 thesubalgebra F q p N z G ` q “ à λ P P ` l ´ λ b V λ Ă F q p G ` q , which we regard as an algebra object of Rep q p G q b Rep q p T q with acting U q p g q in the secondtensor factor V λ , and U q p t q acting in l ´ λ . Then we have an isomorphism of algebras inRep q p G q b Rep q p T q F q p N z G ` q ÝÑ End G ˆ T p B q , v ˚´ λ b v ÞÝÑ p χ _ λ b v ˚´ λ q b v b χ ´ λ , where χ _ λ , χ ´ λ are any elements of C ˚´ λ , C ´ λ respectively normalized so that x χ _ λ , χ ´ λ y “ G ˆ T p B q with F q p N z G q as analgebra in Rep q p G ˆ T q , and in the sequel we will always choose to do so in order to avoidextraneous powers of q appearing in various explicit formulas. Remark 3.40.
A similar computation will be used to calculate r L in Theorem 3.45. Proposition 3.41.
The unit object B P Rep q p B q is a Rep q p G ˆ T q -generator, in other wordsthe functor Hom G ˆ T p B , ´q “ act R B is conservative. Proof.
It is equivalent to show in that case that act B is dominant, i.e. that for anynon-zero U q p b q -module M , there exists some V P Rep q p G q and χ P Rep q p T q such thatHom p i ˚ V b π ˚ χ, M q ‰
0. Clearly we may further assume (by passing to a submodule) that M is cyclic on some weight vector m ν , so that being finite-dimensional, it is a quotient of U q p b q b U q p t q C ν , for some T -weight ν . We may choose V “ V λ for λ suitably large that theannihilator of V , intersected with the subalgebra U q p n q , contains the annihilator in U q p n q of M . Then there is a unique U q p n q -module map from V λ Ñ M sending the highest weightvector v λ to m ν and the only obstruction to this being a U q p b q -map is that λ ‰ ν . Hence weobtain a non-zero map i ˚ V λ b π ˚ C ν ´ λ Ñ M . (cid:3) Definition 3.42. An F q p N z G q -module M is torsion if, for every m P M , there exists a λ such that µ ě λ implies that V µ m “
0. We denote the collection of torsion modules by Tors.A module M is torsion-free if it does not contain any non-zero torsion submodules. UANTUM DECORATED CHARACTER STACKS 33
We note that the sum Tors p M q of all torsion submodules of any module M is again asubmodule of M , and that the quotient of M by this submodule is torsion free. Moreover,Tors is a Serre subcategory, meaning that it is closed under formation of long exact sequences,and the quotient category is again abelian. Lemma 3.43.
We have the following characterization of U q p g q b U q p t q -invariant ideals in F q p N z G q .(1) For a dominant weight λ , the subspace I λ “ ‘ µ ě λ p V µ b C ´ µ q is an invariant ideal.(2) Every invariant ideal in I is a finite sum I “ ř i I λ i of such ideals. Proof.
Claim (1) is clear, since multiplication in F q p N z G q only increases weights in theordering. For Claim (2), we note that every U q p g q b U q p t q -subspace of F q p N z G q is a directsum of its full isotypic components, because F q p N z G q is multiplicity-free. Moreover, we recallthat the multiplication in F q p N z G q is such that m p V λ b V µ q “ V λ ` µ , so that once V λ occursin the ideal, all its dominant translates do as well. (cid:3) Corollary 3.44.
An object M of F q p N z G q´ mod G ˆ T is torsion-free if, and only if all itscompact subobjects have zero annihilator ideals. Proof.
If we suppose M is torsion, clearly all its compact subobjects have non-trivialannihilator ideals. Conversely, as soon as the annihilator ideal of any compact subobject isnon-empty, then it is of the form I “ ř i I λ i described in Lemma 3.43, which implies that it istorsion, since we may take λ “ λ i for any i in the definition of a torsion module. (cid:3) Theorem 3.45.
The comparison functor
Hom G ˆ T p B , ´q : Rep q p B q ÝÑ F q p N z G q´ mod G ˆ T V ÞÝÑ p V b F q p G ˆ T qq U q p b q , defines a fully faithful embedding, realizing Rep q p B q as the open subcategory Tors K . Proof.
Thanks to Proposition 3.41, we may apply Theorem 3.16 to conclude that Hom G ˆ T p B , ´q yields a fully faithful embedding of Rep q p B q as an open subcategory of F q p N z G q´ mod G ˆ T .So by Proposition 3.19, all that remains is to identify the kernel of its left adjoint r L with Tors.In complete analogy with the classical parabolic induction functor from Example 3.32, thisleft adjoint is given by r L : F q p N z G q´ mod G ˆ T ÝÑ Rep q B, M
ÞÝÑ M {p ker ε q M, where ker ε is the 2-sided ideal in F q p N z G q given by the kernel of the evaluation morphism ε : F q p N z G q Ñ B , and the U q p b q -action on M {p ker ε q M is inherited from that of U q p g qb U q p t q on M .Now suppose that M is an object of Tors.Recall the principal minors t f λ u λ P Λ ` defined in Remark 3.35, and observe that f λ P F q p N z G q ,while ε p f λ q “ λ P Λ ` . So for each m P M , choosing λ sufficiently large that f λ ¨ m “ m “ ¨ m “ p ´ f λ q m P p ker ε q M , and thus r L p M q “ M is torsion-free, then M { ker εM ‰
0. Forthis, let us first suppose that M is finitely generated as an F q p N z G q –module. Given a weight λ , denote by M r λ s the corresponding weight space of M with respect to the U q p t q -action. Let λ be the highest weight such that M r λ s ‰
0. Since for any weight λ ker p ε q M r λ s Ď ‘ µ ď λ M r µ s , D G D T D T D T D T D T Figure 7.
A gated pentagon D with a depiction of the action by disk insertions.the condition M “ ker p ε q M implies M r λ s Ď ker p ε q M r λ s . Now let t m , . . . , m n u be a basisof M r λ s . Then any element m P M r λ s can be written as m “ ÿ i h i m i (3.6)for some h i P ker ε . Note that by comparing the T -weights of both sides, we may assumethat the h i are elements of the (commutative) polynomial ring C p q qr Λ ` s generated by theprincipal minors f λ . We may thus make a similar argument to that used to prove the classicalNakayama lemma: expand m i “ ÿ j f ij m j (3.7)and form the n ˆ n –matrix δ ij ´ f ij with entries in C p q qr Λ ` s . Multiplying Eq. (3.7) throughby its adjugate matrix, we find that det p δ ij ´ f ij q annihilates all m i , and is of the form1 ` ker ε . The annihilator of the m i is thus nonzero, which by Proposition 3.44 contradictsthe assumption that M is torsion-free.To treat the general case, we note that r L is exact on torsion free modules. Hence, if r L p M q “
0, then for any finitely generated submodule M ã Ñ M we must also have r L p M q “ M is torsion. Hence M is torsion, as required. (cid:3) Monadic reconstruction: Z p D n q and F q p Conf frn q .Definition 3.46. We define the braided tensor category Rep q p G ˆ T n q byRep q p G ˆ T n q : “ Rep q p G q b Rep q p T q b n . Definition 3.47.
We endow the categories Z p D n q with an action of the monoidal categoryRep q p G ˆ T n q , by choosing a single interval in B S X S G , and one in each connected componentof B S X S T , as indicated in Fig. 7. We let Dist denote the distinguished object of the category Z p D n q . Definition 3.48.
For n ě
1, the quantum coordinate algebra of n framed flags is the n -fold braided tensor product, F q p Conf frn q : “ F q p N z G q b ¨ ¨ ¨ b F q p N z G q . UANTUM DECORATED CHARACTER STACKS 35
By definition, this means that as an object it is the tensor product in Rep q p G ˆ T n q , andthat multiplication within each factor is given by the algebra structure of F q p N z G q and thatrelations between the factors are given by b p j q a p i q “ ´ R p q á a p i q ¯ ´ R p q á b p j q ¯ , (3.8)where 1 ď i ă j ď n and x p i q denotes the element x placed in the i -th braided tensor factor,where R denotes the universal R -matrix of U q p g q . We note that each subalgebra F q p N z G q occupies a distinct Rep q p T q factor, while sharing the same Rep q p G q factor, hence the braidedtensor product involves only the universal R -matrix for U q p g q . Theorem 3.49.
Let
Dist D n P Z p D n q denote the distinguished object. We have an isomorphism End G ˆ T n p Dist D n q – F q p Conf frn q . Proof.
The case n “ n ą D n “ D \ D G ¨ ¨ ¨ \ D G D , hence we obtain anequivalence of categories, Z p D n q » Rep q p B q b Rep q p G q ¨ ¨ ¨ b Rep q p G q Rep q p B q , and therefore applying [BBJ18a][Corollary 4.13], we have an isomorphism,End G ˆ T n p Dist D n q » End G ˆ T p B q r b ¨ ¨ ¨ r b End G ˆ T p B q , where r b denotes the braided tensor product of algebras in Rep q p G q , as required. (cid:3) Definition 3.50.
For n ě
1, we denote by Tors n Ă F q p Conf frn q´ mod G ˆ T n the full Serresubcategory consisting of modules whose restriction to some F q p Conf fr q subfactor is torsion. Proposition 3.51.
For any n ě , the functor Hom G ˆ T n p Dist D n , ´q induces an equivalenceof categories, Z p D n q » Tors K n Ă F q p Conf frn q´ mod G ˆ T n . Proof.
Let r L n denote the left adjoint. By Proposition 3.19 we need to identify the kernelof r L with Tors n . The case n “ n ą
1, we have have theequivalence, Z p D n q » Rep q p B q b Rep q p G q ¨ ¨ ¨ b Rep q p G q Rep q p B q . Under this equivalence r L identifies with the functor, r L b Rep q p G q ¨ ¨ ¨ b Rep q p G q r L . The canonicalfunctor from r L b ¨ ¨ ¨ b r L to the relative tensor product is conservative, hence we haveker p r L n q “ ker p r L q b Rep q p G q ¨ ¨ ¨ b Rep q p G q ker p r L q “ Tors n . (cid:3) Further computations of F q p Conf frn q for G “ SL . For G “ SL , the algebra F q p G q is generated by matrix coefficients of the first fundamental representation, and thus can bewritten as F q p G q » C p q q x a ij y with i, j “ , , where a a “ qa a , a a “ qa a , a a “ a a , and a a ´ qa a “ . Spelling out the above equality, and setting x “ a and x “ a in each factor, we arrive atthe following Proposition. Proposition 3.52.
The algebra F q p Conf frn q is an associative algebra over C p q q with generators x p i q , x p i q , ď i ď n, and relations x p i q x p i q “ qx p i q x p i q for all ď i ď n, and x p j q x p i q “ q ´ x p i q x p j q ,x p j q x p i q “ q x p i q x p j q ,x p j q x p i q “ q ´ x p i q x p j q ` p q ´ ´ q q x p i q x p j q ¯ , (3.9) x p j q x p i q “ q ´ x p i q x p j q . for all ď i ă j ď n . All the above identities in Proposition 3.52 follow from viewing functions on N z G as a G -representation, and using the R -matrix to braid the factors in F q pp N z G q n q before multiplying. Definition 3.53.
For any pair 1 ď i, j ď n such that i ‰ j , we define elements D ij P F q p Conf frn q by D ij “ x p i q x p j q ´ qx p i q x p j q . Lemma 3.54.
For ď i ă j ď n we have D ji “ ´ q D ij . Proof.
The proof follows immediately from relation (3.9) in Proposition 3.52: D ji “ x p j q x p i q ´ qx p j q x p i q “ q x p i q x p j q ´ q ´ x p i q x p j q ` p q ´ ´ q q x p i q x p j q ¯ “ ´ q D ij . (cid:3) Proposition 3.55.
Elements D ij are invariant with respect to the (left coregular) U q p sl q -action, that is D ij P F q p Conf frn q U q p sl q . Moreover, for all ď i ă j ď n , and a “ , , theysatisfy commutation relations D ij x p k q a “ $’&’% q x p i q a D ij , if k “ i,q ´ x p j q a D ij , if k “ j,x p k q a D ij , if k ă i or k ą j, as well as D ij x p k q a “ q ´ x p j q a D ik ` q x p i q a D kj ,x p k q a D ij “ q D ik x p j q a ` q ´ D kj x p i q a UANTUM DECORATED CHARACTER STACKS 37 if i ă k ă j . Proof.
Note that we have E á x “ , K á x “ qx , F á x “ x ,E á x “ x , K á x “ q ´ x , F á x “ . The proof of the invariance of degree 2 element D ij then follows from a direct calculationusing the module algebra structure. For example, we have E á D ij “ ´ p E á x p i q q x p j q ` p K á x p i q qp E á x p j q q ¯ ´ q ´ p E á x p i q q x p j q ` p K á x p i q qp E á x p j q q ¯ “ qx p i q x p j q ´ qx p i q x p j q “ . As for the commutation relations, we shall only prove the top equality here, with the othercases being proved in a similar fashion. Substituting a “ , D ij x p i q “ x p i q x p j q x p i q ´ qx p i q x p j q x p i q “ q x p i q x p i q x p j q ´ q x p i q x p i q x p j q “ q x p i q D ij and D ij x p i q “ x p i q x p j q x p i q ´ qx p i q x p j q x p i q “ q x p i q ´ x p i q x p j q ` p q ´ ´ q q x p i q x p j q ¯ ´ q ´ x p i q x p i q x p j q “ q x p i q ´ x p i q x p j q ´q x p i q x p j q ¯ “ q x p i q D ij . (cid:3) As an immediate corollary of Proposition 3.55, we obtain the following commutationrelations between the D ij . Corollary 3.56.
For i ă k ă j , we have D ik D kj “ q ´ D kj D ik ,D kj D ij “ q ´ D ij D kj ,D ij D ik “ q ´ D ik D ij . We also obtain the following exchange relation:
Corollary 3.57.
For i ă k ă j ă l , we have D ij D kl “ q ´ D ik D jl ` q D kj D il . Next, we will show that certain sets of functions D ij are multiplicative Ore sets, which willallow us to localize the ring F q p Conf frn q .We will say that an n -gon D n is based if its vertices are labelled 1 to n in a counterclockwiseorder. For any 1 ď k ă n , a step k diagonal e ij in a based n -gon D n is a segment connectingvertices i and j with j ´ i “ k . In particular, all step 1 diagonals are sides of D n , as well as the only step p n ´ q diagonal e n . In what follows, diagonals of step 2 will be referred to as short diagonals . We say that diagonals e ij and e kl are crossing if either of the following twoinequalities holds: i ă k ă j ă l or k ă i ă l ă j. A triangulation of D n is a maximal collection of its non-crossing diagonals. Note that everytriangulation contains all sides of D n and at least one short diagonal. We will call anycollection, not necessarily maximal, of non-crossing diagonals of D n its partial triangulation. To an algebra F q p Conf frn q let us associate an n -gon D n , whose vertices represent tensorfactors of F q p Conf frn q and are labelled accordingly. A partial triangulation ∆ of D n gives riseto the multiplicative set S ∆ Ă F q p Conf frn q generated by elements D ij such that e ij P ∆. Notethat by Corollary 3.56, all elements of S ∆ skew-commute. Lemma 3.58.
For any element r P F q p Conf frn q and any triangulation ∆ of D n , there existelements s r , s r P S ∆ such that both products s r r and rs r can be expressed as sums of monomials in x p q a , x p n q a , and D ij P S ∆ . Proof.
We shall prove the Lemma by induction on n . For n “ r P F q p Conf frn q can be uniquely written as a sum r “ m ÿ (cid:96) “ c (cid:96) ´ x p q ¯ α (cid:96), ´ x p q ¯ β (cid:96), . . . ´ x p n q ¯ α (cid:96),n ´ x p n q ¯ β (cid:96),n , where c (cid:96) P C p q q and m is the number of monomials in the expression. Let us set d i “ max (cid:96) p α (cid:96),i ` β (cid:96),i q . Now, consider a triangulation ∆ of D n and any of its short diagonals e “ e i ´ ,i ` P ∆. ByProposition 3.55 and Corollary 3.56, we have relations D e x p i q a “ q ´ x p i ` q a D i ´ ,i ` q x p i ´ q a D i,i ` ,x p i q a D e “ q D i ´ ,i x p i ` q a ` q ´ D i,i ` x p i ´ q a and know that D e skew-commutes with all monomials in the right hand side of the aboveequalities. This in turn implies that the products D d i e ¨ r and r ¨ D d i e can be expressed as sums of monomials in elements D i ´ ,i , D i,i ` , and x p j q a with j ‰ i .Now, let D n ´ be the p n ´ q -gon obtained from D n by erasing the sides e i ´ ,i and e i,i ` .Choosing a short diagonal e “ e j ´ ,j ` of D n ´ and repeating the above reasoning, we derivethat the products D d j e D d i e ¨ r and r ¨ D d i e D d j e can be expressed as sums of monomials in elements D i ´ ,i , D i,i ` , D j ´ ,j , D j,j ` , and x p k q a with k ‰ i, j . The rest of the proof is done by induction on the number n of sides in D n . (cid:3) Corollary 3.59.
For any triangulation ∆ of D n , the collection S ∆ forms a multiplicative(left and right) Ore set in the ring R “ F q p Conf frn q : for all r P F q p Conf frn q and s P S ∆ onehas S ∆ r X Rs ‰ H and rS ∆ X sR ‰ H . UANTUM DECORATED CHARACTER STACKS 39
Proof.
By Lemma 3.58, for any r P R there exist elements s r , s r P S ∆ so that the products s r r and rs r can be expressed as sums of monomials in x p q a , x p n q a , and D ij P S ∆ . Since elementsof S ∆ skew-commute with each other, as well as with x p q a and x p n q a , we conclude that ss r r “ r s and rs r s “ sr for some r , r P R . (cid:3) Definition 3.60.
Let ∆ be a partial triangulation of D n . We will denote by F q p Conf frn qr S ´ s the Ore localization of F q p Conf frn q at the multiplicative Ore set S ∆ . In particular, we willwrite F q p Conf frn qr e ij ´ s for F q p Conf frn qr D ´ ij s . Lemma 3.61.
For n ě and ď i ă j ď n , there is an injective map of U q p g q -modulealgebras η ij : F q p G q ÝÑ F q p Conf frn qr e ij ´ s defined by a ÞÝÑ x p i q , a ÞÝÑ x p j q D ´ ij ,a ÞÝÑ x p i q , a ÞÝÑ x p j q D ´ ij . Proof.
The statement of the Lemma is easily verified with the help of relations establishedin Proposition 3.52 and Proposition 3.55. (cid:3)
Cluster charts on Z p D n q for G “ SL . We have the algebra F q p Conf frn q along withits localization F q p Conf frn qr S ´ s . We would like to relate the localization with Z p D n q . Byconstruction the two inclusions indicated by solid arrows below define open subcategories: wecollect the embedding functors and their left adjoints in the diagram below. Z p D n q F q p Conf frn qr S ´ s´ mod G ˆ T n F q p Conf frn q´ mod G ˆ T n We define the dotted arrows as the evident compositions. In this section we will show that thedotted arrows realize F q p Conf frn qr S ´ s´ mod G ˆ T n as an open subcategory of Z p D n q . Havingalready identified Z p D n q as the subcategory of torsion-free modules, it remains only to showthat localizing to F q p Conf frn qr S ´ s kills all torsion modules. Theorem 3.62.
The dotted arrows realize F q p Conf frn qr S ´ s´ mod G ˆ T n as an open subcategoryof Z p D n q . Proof.
Let us identify the weight lattice of SL with Z via ω ÞÑ , α ÞÑ
2. As an algebraobject in Rep q p G ˆ T n q , F q p Conf frn q is graded by the torus T n ; let us write F q p Conf frn q p i ; k q for the set of elements of weight ě k with respect to the i th T -factor, and F q p Conf frn q k(cid:15) i forthe subset of elements whose weight with respect to T n is the vector k(cid:15) i with a k in position i and zeros elsewhere. Then it is evident from the homogeneity of the relations (3.9) that wehave F q p Conf frn q p i ; k q “ F q p Conf frn q F q p Conf frn q k(cid:15) i . (3.10)Suppose some F q p Conf frn q -module M admits a non-zero morphism from a torsion module.Then there must exist an element m P M , an index 1 ď k ď n , and a natural number N suchthat r ě N ùñ F q p Conf frn q r(cid:15) k m “ . Now any Z ij such that k P t i, j u is an element of F q p Conf frn q k ;1 . So by Eq. (3.10), wehave Z rij P F q p Conf frn q k ; r “ F q p Conf frn q F q p Conf frn q r(cid:15) k . Therefore p Z ij q r m “
0, whichshows that the element Z ij does not act invertibly on such a module M . Thus we have F q p Conf frn qr S ´ s´ mod G ˆ T n Ă Tors K . (cid:3) Theorem 3.63.
For any partial triangulation ∆ ‰ H of a based n -gon associated to F q p Conf frn q , the functor of taking U q p g q -invariants, G : F q p Conf frn q´ mod G ˆ T n ÝÑ F q p Conf frn q U q p g q ´ mod T n , restricts to an equivalence of categories, G : F q p Conf frn qr S ´ s´ mod G ˆ T n ÝÑ F q p Conf frn qr S ´ s U q p g q ´ mod T n . Proof.
Because the category Rep q p G q is semisimple, the functor G of taking U q p g q -invariantsis exact. Thus to prove that it is an equivalence, it suffices to show that it is also conservative.Since the partial triangulation ∆ is not empty, there is an edge e ij P ∆ and hence an injectivehomomorpshim η ij : F q p G q Ñ F q p Conf frn qr S ´ s by Lemma 3.61. This in turn allows us toinclude the functor G into the commutative diagram F q p Conf frn qr S ´ s´ mod G ˆ T n G (cid:47) (cid:47) F (cid:15) (cid:15) F q p Conf frn qr S ´ s U q p g q ´ mod T n F (cid:15) (cid:15) F q p G q´ mod G ˆ T n G (cid:47) (cid:47) Rep q p T n q where F and F are forgetful functors, and G is again the functor of taking U q p g q -invariants.Now, it is enough to show that the composition G ˝ F is conservative. Being a forgetfulfunctor, F is obviously conservative, while G is in fact an equivalence with inverse given by V ÞÑ F q p G q b V . (cid:3) Charts and flips on Z p S q via excision In this section we will compute with an arbitrary simple decorated surface S . We willdescribe a special family of open subcategories of the category Z p S q , which we regard as opensubvarieties – charts – on the quantum decorated character stack of S . We have one suchchart for each isotopy class of triangulation of the S , and each chart comes equipped with adistinguished compact projective generator Dist ∆ , and an identification as the category ofmodules for a quantum torus, resembling the notion of a cluster chart.The basic mechanism to obtain these charts is to start from the monadic descriptionof Z p D q obtained in the previous section, and then apply excision with respect to gluingtriangles along a common digon D . The charts so constructed admit pairwise transitionmaps resembling those in the quantum A -variety variety of Fock–Goncharov – we call thewhole structure a quantum p A -variety , and we compare it to the quantum cluster A -variety of UANTUM DECORATED CHARACTER STACKS 41
Fock–Goncharov. Taking T -invariants yields a version of the quantum cluster ensemble map,giving a reconstruction of cluster X -varieties.4.1. Opening T -gates. If S is a decorated surface with a collection G of gates, recall thatthe category Z p S q inherits the structure of a module category for the braided tensor categoryRep q p G q “ Rep q p G m b T n q , where m and n denote the number of G - and T -gates in G . Givensome object X P S – typically in examples X will be some Dist ∆ – different choices of G clearly produce distinct internal endomorphism algebras End G p X q .However, the category End G p X q´ mod G of End G p X q -modules in Rep q p G q does not changewhen we open gates. Lemma 4.1.
Let G be some configuration of T -gates, and let G be obtained from G byreplacing any T -gate with a pair of adjacent gates. Then the functor of taking invariants ateither new gate yields an equivalence of categories: End G p Dist S q´ mod G » End G p Dist S q´ mod G . Proof.
Indeed, in this case the G -module category End G p Dist S q´ mod Rep q p G q is simply thepullback of End G p Dist S q´ mod Rep q p G q along the tensor functor Rep q p T q b Rep q p T q Ñ Rep q p T q . (cid:3) The specification of T -gates, as well as the presentation for the resulting algebras of internalendomorphisms will be encoded combinatorially in the data of a fencing graph associatedto the triangulation. Definition 4.2.
Let ∆ be a triangulation of a simple decorated surface. Then the boundaryof the G -region of every digon or triangle in ∆ consists of interlacing short and long , wherethe former lie in S T , the latter in S G .Thus each triangle or digon gives rise to a hexagon or rectangle respectively with alternatinglong and short edges. Definition 4.3.
The fencing graph Γ ∆ has one vertex for each corner of a hexagon appearingin ∆. The edge set consists of a directed arrow connecting opposite vertices of each short edgeof either a hexagon or a rectangle, with the direction being compatible with the clockwiseorientation within T -region, and a bivalent edge connecting the opposite pair of vertices ofeach long edge.An example of a fencing graph for the punctured triangle D ˝ is illustrated in Fig. 8. Clearly,there are 6 h vertices in Γ ∆ , where h is the number of triangles in ∆. We associate to thefencing graph Γ ∆ a collection of T -gates in S by declaring that each vertex of Γ ∆ lies in aunique T -gate. Thus a notched triangulation ∆ ‚ determines in each T -region a distinguishedvertex of the fencing graph Γ ∆ , for convenience we assume that as we traverse the boundaryof any T -disk in a counterclockwise direction, we encounter the distinguished vertex first.It also induces a total ordering on the edge-ends incident to vertices at each T -region: wenumber edge-ends as we traverse the T -region clockwise, starting with the one incident to thedistinguished vertex.This in turn determines a notched ideal triangulation , where we identif all vertices ineach T -region of the fencing graph, collapse parallel long edges, and collapse all short edgesbut one whose head is the distinguished vertex. The remaining short edge becomes a loop,which we call a notch ; this defines a total ordering on the edge-ends of the notched idealtriangulation, see Fig. 8. (cid:134) (cid:134) Figure 8.
Presentation of a punctured triangle D ˝ as in Eq. (1.1) and Eq. (4.1),the corresponding fencing graph with a distinguished vertex, and the notchedideal triangulation with the total numbering on the edge-ends adjacent to thepuncture. The location of the distinguished gate in D ˝ , and the distinguishedvertex of the fencing graph, are marked with .We will be considering the following configurations of gates associated to a notchedtriangulation ∆ ‚ : ‚ T Γ has a single gate located at each vertex of the fencing graph. ‚ T P contains only the distinguished gate at each puncture. ‚ T P c contains only the non-distguished gates at each puncture. ‚ T M contains only the gates at each marked point.Hence, we have a decomposition T Γ “ T P c b T P b T M Remark 4.4.
Let us discuss the rationale for introducing additional T -gates. We will seebelow that opening gates produces quantum tori r ζ p ∆ ‚ q with more generators than the tori ζ p ∆ ‚ q , in which we are ultimately interested. However, the larger tori r ζ p ∆ ‚ q have simplercommutation relations which are expressed more locally in the triangulation and hence easierto compute, as in Propositions 4.7 and 4.6 below. So we compute with the algebra r ζ p ∆ ‚ q while performing excision, and close the additional gates only at the end.4.2. Charts on Z p D q and Z p D q . Let us first consider the two cases when S is a triangle ordigon. Recall from Theorem 3.62 the open subcategories Z p D q Ă Z p D q and Z p D q Ă Z p D q defined, respectively, by the condition that elements of S D and S D act invertibly. We considerthe restriction of the distinguished object to each subcategory, and define ζ p D q : “ End T p Dist D q – F q ´ Conf fr r D ´ s ¯ U q p g q ,ζ p D q : “ End T p Dist D q – F q ´ Conf fr r D ´ s ¯ U q p g q , where the T -actions are induced by introducing one T -gate in each T -region. By Theorem 3.63,we have equivalences of categories Z p D q » ζ p D q´ mod T , Z p D q » ζ p D q´ mod T . It is evident that we have ζ p D q » C p q q @ D ˘ D , and the following Lemma shows that thealgebra ζ p D q is also a quantum torus. UANTUM DECORATED CHARACTER STACKS 43
Lemma 4.5.
The algebra ζ p D q is generated by the elements D ˘ ij for ď i ă j ď given inthe Definition 3.53. Proof.
The algebra F q p Conf fr q U q p g q is the direct sum of its weight subspaces with respect to U q p t q b : F q p Conf fr q U q p g q “ à l,m,n ě ´ F q p Conf fr q U q p g q ¯ l,m,n , and each of these subspaces is at either 1-dimensional if l ` m ` n is even and there exists aEuclidean triangle with side lengths l, m, n , or zero-dimensional otherwise. But it is straight-forward to see that the submodule of Z generated by the weights p , , q , p , , q , p , , q ofthe D ij is exactly the lattice of all vectors with even coordinate sum. Thus for each triple p l, m, n q such that the corresponding subspace of F q p Conf fr q U q p g q is non-zero, there exists atriple of integers n , n , n P Z such that D n D n D n P ´ F q p Conf fr q U q p g q ¯ α,β,γ . (cid:3) Opening an additional gate at each T -region equips Z p D q and Z p D q with T - and T -actions, respectively, over which the restricted distinguished objects are still relative generators.Thus writing r ζ p D q : “ End T ´ Dist D ¯ , r ζ p D q : “ End T ´ Dist D ¯ , we obtain equivalences of categories Z p D q » r ζ p D q´ mod T , Z p D q » r ζ p D q´ mod T . Lemma 4.6.
The algebra r ζ p D q is generated by elements A , A in correspondence with thelong edges of the fencing graph for D , as well as generators a , a in correspondence with itsshort edges. These generators are subject to the relations r A , A s “ r a , a s “ , a i A j “ q A j a i , and a A “ a A . Proof.
The Lemma is a special case of Example 3.38, where we open one additional gatein each T -region. Introducing the weights ε “ p , q , ε “ p , q for the Rep q p T q –action on ζ p D q and setting C λ,µ “ C λ b C µ P Rep q p T q b Rep q p T q , we take the generators A i , a i to be A “ D b χ ε ,ε , a “ b χ ε , ´ ε ,A “ D b χ ε ,ε , a “ b χ ε , ´ ε . The asserted relations then follow from the general multiplication rule (3.5). (cid:3)
Similarly, in the case of D we have Lemma 4.7.
The algebra r ζ p D q is a quantum torus with generators A , A , A in corre-spondence with the long edges of the fencing graph for D , as well as generators a , a , a incorrespondence with its short edges. The relations among these generators are r A ij , A kl s “ r a i , a j s “ and a k A ij “ q δik ` δkj A ij a k . Proof.
The proof is identical to that of Lemma 4.6. The generators are given by A “ D b χ ε ,ε , A “ D b χ ε ,ε , A “ D b χ ε ,ε , and a k “ b χ ε k , ´ ε k , for k “ , , , with the asserted relations between them following immediately from formula (3.5). (cid:3) The above computations were all for G “ SL . However, we have the following paralleldescriptions for G “ PGL . The proof in this case is identical except that one requires allrepresentations which appear to have exclusively even weights. Corollary 4.8.
We have the following: ‚ Each algebra ζ PGL p D q is a quantum sub-torus of ζ SL p D q generated by D ˘ . ‚ Each algebra ζ PGL p D q is a quantum sub-torus of ζ SL p D q generated by D ˘ , D ˘ , D ˘ , p D D D q ˘ . ‚ Each algebra r ζ PGL p D q is a quantum sub-torus of ζ SL p D q generated by A ˘ , A ˘ , a ˘ , a ˘ . ‚ Each algebra r ζ PGL p D q is a quantum sub-torus of ζ SL p D q generated by A ˘ , A ˘ , A ˘ , p A A A a a a q ˘ , a ˘ , a ˘ , a ˘ . . Remark 4.9.
We remark that a monomial subalgebra of a quantum torus is necessarily aquantum torus, so that even though there are relations above the generators listed above onecan always (non-canonically) choose free generators. For instance, in the second list one maytake D , D , D D D . Remark 4.10.
Note that the orientation of D defines a cyclic ordering on its T -regions. Inwhat follows, we regard the generator A ij as being in correspondence with the long edge of thefencing graph Γ D connecting the T -regions labelled i and j , and the generator a k as being incorrespondence with the short edge of Γ D sitting on the boundary of the k -th T -region.4.3. Charts on Z p S q from triangulations. Suppose now that ∆ is a triangulation of ageneral decorated surface S consisting of t triangles and l digons. To organize computations,we shall fix a framing of S , in which all triangles are drawn in the plane of the blackboardwith their long edges parallel to its x -axis. Each digon we represent as a ribbon connecting apair of long edges of the triangles. In particular, such a framing of S fixes for each long edgeof the fencing graph Γ ∆ an ordering on the vertices at its endpoints. An example of such aframing in the case that S is a punctured torus is drawn in Fig. 9. We wish to emphasizethat this additional choice of a framing is immaterial to the determination of the charts wewill construct, but is merely convenient for giving presentations, and formulas for mutations.See Section 4.9 for more details.The triangulation ∆ is a presentation of S as S “ p D q \ t ğ p D q \ (cid:96) p D q \ (cid:96) , where the 2 l digons over which we glue are obtained as small neighborhoods of the longedges of the fencing graph Γ ∆ as depicted in Fig. 9. Applying excision, we thus obtain anequivalence of categories Z p S q » Z p D q t ò Z p D q l Z p D q l (4.1) UANTUM DECORATED CHARACTER STACKS 452 6 4 5 1 234 1 2 3 4 5 6
Figure 9.
The once-punctured torus with its fencing graph overlaid, and its2-framing as presented by a pair of jellyfish.which allows us to define a open subcategory Z p ∆ q of Z p S q by Z p ∆ q : “ Z p D q t ò Z p D q l Z p D q l . (4.2)By construction, r ζ p D q b l is a commutative algebra object in the braided tensor category T Γ .Moreover, in view of Lemmas 4.7 and 4.6 we have equivalences Z p D q t » r ζ p D q b t ´ mod T Γ , Z p D q l » r ζ p D q b l ´ mod T Γ , where r ζ p D q b t and r ζ p D q b l are both r ζ p D q b l –algebras. We thus find ourselves in the setting ofLemma 1.17, which we may apply to obtain the following description of the open subcategorycorresponding to the triangulation ∆: Z p ∆ q » r ζ p ∆ q´ mod T Γ , where r ζ p ∆ q : “ End T ∆ p Dist ∆ q – r ζ p D q b t b ζ p D q b l r ζ p D q b l . The quantum torus r ζ p ∆ q can be described explicitly with the help of the relations fromLemmas 4.7 and 4.6. Let us write E p Γ ∆ q for the set of edges of the fencing graph Γ ∆ , so that E p Γ ∆ q “ E l p Γ ∆ q \ E s p Γ ∆ q , where E l p Γ ∆ q and E s p Γ ∆ q denote the subsets of long and short edges respectively. Recallthat the union of all short edges coincides with the set of walls C between G and T -regions.Let us now orient the short edges so that they travel around T -regions in a clockwise manner.Given a short edge e P E s p Γ ∆ q , we define its head h p e q and its tail t p e q , to be its endpointsso that e is oriented from t p e q to h p e q . Proposition 4.11.
The algebra object r ζ p ∆ q in the category T Γ is a quantum torus withgenerators A ˘ (cid:96) , (cid:96) P E l p Γ ∆ q , a ˘ e , e P E s p Γ ∆ q . These generators are subject to the following fencing graph relations which hold for all long edges (cid:96), (cid:96) and short edges e, e : A (cid:96) A (cid:96) “ A (cid:96) A (cid:96) , a A “ a A , (4.3) if a , a , A , A are the four edges in a digon as in Lemma 4.6, a e A (cid:96) “ q A (cid:96) a e if (cid:96) X e ‰ H ,A (cid:96) a e otherwise ; ,a e a e “ q a e a e if h p e q “ t p e q and t p e q ‰ h p e q ,a e a e otherwise . , The T ∆ -equivariant structure is as follows: wt i p A (cid:96) q “ if i P (cid:96), if i R (cid:96), wt i p a e q “ $’&’% if i “ t p e q , ´ if i “ h p e q , otherwise , where wt i p x q P Λ p T q indicates weight of an element x P r ζ p ∆ q with respect to the torus at the i -th vertex in Γ ∆ . Corollary 4.12.
The algebra object r ζ PGL p ∆ q is the subalgebra of r ζ SL p ∆ q generated by p a i q ˘ for each short edge, A ˘ (cid:96) for each long edge, and p A (cid:96) A (cid:96) A (cid:96) a a a q ˘ for each triangle. Closing T -gates.Definition 4.13. If ∆ ‚ is a notched triangulation of a decorated surface S , we define thequantum torus ζ p ∆ ‚ q to be the subalgebra of T P c -invariants in r ζ p ∆ q .By Lemma 4.1, the functor of taking T P c invariants is conservative. Hence we have anequivalence of categories, ζ p ∆ ‚ q´ mod T P Y M » r ζ p ∆ q´ mod T Γ . The generators of ζ p ∆ ‚ q are labelled by the edges and the notch of the notched idealtriangulations. We shall now describe these generators and their relations, and express themin terms of generators of r ζ p ∆ q . We will make heavy use of the Weyl ordering on the quantumtorus r ζ p ∆ q . Definition 4.14.
Suppose that ζ is a quantum torus with a basis of generators t X i u . Thenthe Weyl ordering with respect to the generating set p X i q is defined recursively by declaring : X i : “ X i for all i , and if Y and Z are two monomial elements satisfying Y Z “ q r ZY , q ´ r : Y : : Z : “ : Y Z : “ q r : Z : : Y : . Let s be an edge in the notched triangulation ∆ ‚ , and pick a long edge l in the fencinggraph corresponding to s . Then the endpoints of l are vertices w , w of the fencing graphcontained in regions whose distinguished vertices are v and v . Denote by a p l q the set ofshort edges in the minimal path in the fencing graph starting at distinguished vertex v andfollowing the short edges clockwise to the vertex w . Similarly we define a p l q to be the set ofshort edges in the minimal path in the fencing graph starting at the distinguished vertex v and travelling clockwise to the vertex w . Then we define an element Z s of ζ p ∆ ‚ q by Z s : “ : a p l q a p l q A l : . It follows from the relations (4.3) in Proposition 4.11 that the element Z s does not depend onthe choice of long edge l projecting to the edge s in the underlying notched triangulation. UANTUM DECORATED CHARACTER STACKS 47
Given a puncture p P p S , let v P Γ ∆ be the corresponding distinguished vertex, and γ p bethe path which connects v to itself and winds once around the corresponding T -region. Thenwe define another element of ζ p ∆ ‚ q as the Weyl-ordered product α p : “ : ź s P γ p a s : , where the product of short edges is taken in clockwise order starting at any vertex in γ p . Wenow describe the weights and commutation relations between the edge variables Z s . In orderto do this, we will make use of the two edge-ends s , s associated to each edge in a notchedtriangulation. Each edge-end s i has a a boundary vertex a “ B s i . Note that the weights ofthe generators Z s and α p with respect to the i -th factor of T P Y M are as follows:wt i p Z s q “ t edge–ends of s lying in the region i u and wt i p α p q “ . (4.4) Definition 4.15.
Let s be an edge-end in a notched triangulation ∆ ‚ , and p “ B s be thevertex at which s terminates. We write | s | for the number of s in the total ordering on the setof edge-ends terminating at the vertex p defined by the distinguished gate.For a pair of edge-ends s and s we now define ω p s , s q “ $’&’% ´ p : “ B s “ B s and | s | ă | s | , p : “ B s “ B s and | s | ă | s | , . Now let s, t P ∆ ‚ be a pair of edges with edge-ends s , s and t , t respectively. We define ω p s, t q by the following formula: ω p s, t q “ ÿ i,j “ ω p s i , t j q . Then from the presentation of r ζ p ∆ q described in Proposition 4.11 we deduce: Theorem 4.16.
For any notching ∆ ‚ of a triangulation ∆ , we have an equivalence ofcategories Z p ∆ q » ζ p ∆ ‚ q´ mod T P Y M . The algebra ζ p ∆ ‚ q is a quantum torus with generators Z ˘ s labelled by the edges s of thenotched ideal triangulation ∆ ‚ , and by an invertible generator α p for each puncture p of S .The relations between these generators are as follows: ‚ for a pair of edges s, t P ∆ ‚ we have Z s Z t “ q ω p s,t q Z t Z s ; ‚ for any puncture p and an edge s with vertices B s , B s we have α p Z s “ q δ p, B s ` δ p, B s Z s α p . Recall now the subalgebra χ p ∆ q of ζ p ∆ ‚ q obtained by closing the distinguished gate ateach puncture: χ p ∆ q “ ζ p ∆ ‚ q T P . Corollary 4.17.
Suppose that either G “ SL and we have at least one marked point, or that G “ PGL . Then the functor of taking T P invariants defines an equivalence, Z p ∆ q » χ p ∆ q´ mod T M . Proof.
First we treat the case G “ SL . Since the functor of taking T P -invariants is exact,we need only show that it is conservative, i.e. that no object is sent to zero, equivalentlythat for any Λ p T P Y M q -graded ζ p ∆ ‚ q -module, the degree-zero subspace with respect to theΛ p T P q -grading is non-zero. For this, simply note that the invertible elements D ij emanatingfrom any puncture have weight 1 at the puncture. Hence, taking ratios D ij D ´ jk for anytriangle with vertices ijk , where i is the puncture, we can shift the degree to vertex k . In thisway, we can move any non-vector to another non-zero vector with degree supported exclusivelyover the non-empty set of marked points.In the case of PGL , we do not need to assume existence of a marked point. In thiscase, we have generators D ij and D ij D jk D kl , and by definition we consider only modulesof even total degree at each gate. For any triangle with vertices ijk , the invertible element p D ij D jk D ki q { D jk has degree 2 at vertex i and degree zero at all other vertices, so it can beused directly to reduce the degree at any vertex i , in particular at the puncture. (cid:3) Corollary 4.18.
Let G “ PGL , and let η p ∆ q “ χ p ∆ q T M . Then we have an equivalence ofcategories, Z PGL p ∆ q » η p ∆ q´ mod . Remark 4.19.
For SL , the functor of taking T -invariants at the final gate is not anequivalence for obvious parity reasons (owing to existence of a center of SL ). However, onecan show that instead of Dist ∆ being a generator of each chart, we have instead that the sumDist ∆ ‘ Dist ∆ r s is a generator, where the latter denotes a parity shift applied to Dist ∆ is agenerator, so that one has an equivalence: Z PGL p ∆ q » p η SL p ∆ q ‘ η SL p ∆ qq ´ mod . Comparison of
PGL and SL charts. Let us consider the commuting diagram ofinclusions of quantum tori, where below each one we recall which T -gates remain open whendefining that algebra. Hence M bijects with the marked points, P bijects with punctures,and r P bijects with the set of incidences p v, T q , where v is a vertex of ∆, T is a triangle of ∆,and v is a vertex of T . η SL p ∆ q χ SL p ∆ q ζ SL p ∆ ‚ q r ζ SL p ∆ q η PGL p ∆ q χ PGL p ∆ q ζ PGL p ∆ ‚ q r ζ PGL p ∆ qH M M Y P M Y r P Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Ð â Ñ Let H p S, X q denote the homology of S relative to X , with Z { Z coefficients. Proposition 4.20.
We have:(1) The algebra η SL p ∆ q is a free module over η PGL p ∆ q of rank | H p S q| .(2) The algebra χ SL p ∆ q is a free module over χ PGL p ∆ q of rank | H p S, M q| .(3) The algebra ζ SL p ∆ ‚ q is a free module over η PGL p ∆ q of rank | H p S, M Y P q| .(4) The algebra r ζ SL p ∆ ‚ q is a free module over r ζ PGL p ∆ q of rank | H p S, M Y r P q| .Moreover in each case, a free basis can be taken consisting of M ε ¨ ¨ ¨ M ε N N , for some monomials M , . . . , M N , with ε i P t , u , and such that M i lies in the SL quantum torus. UANTUM DECORATED CHARACTER STACKS 49
Proof.
We prove (4) first; each of the other statements will follow by computing the appropri-ate subalgebras of invariants. First, an elementary computation with the long exact sequencein relative homology yields | H p S, M Y r P q| “ N , where N “ E ´ F denotes the numberof (long and short) edges in the fencing graph, and F denotes the number of (hexagonal orrectangular) faces. Here, we regard the fencing graph as giving a simplicial decomposition ofthe surface. Then the rank asserted in formula (4) and the required basis in that case followsimmediately from Corollary 4.12: a basis may be given by monomials of degree either one orzero with respect to each (long and short) edge, modulo an equivalence relation identifyingany edge with the product of all other edges on the same face.Each time we close a gate by taking T -invariants at some vertex, the parity requirementthere becomes vacuous, and elements of our basis become redundant. It is then elementary toverify each remaining formula (3), (2), and (1). (cid:3) Comparison of X q ∆ and χ PGL p ∆ q . Let s be an edge in the triangulation ∆, and s , s be its edge ends incident to vertices v and v . Denote by s ` i and s ´ i the edge-ends incident to v i which immediately follow and precede s i in the clockwise order. We then define an element r X s P χ SL p ∆ q as follows: ‚ if s is not a boundary edge and is not the internal edge of a self-folded triangle, r X s “ : Z s ´ Z ´ s ` Z s ´ Z ´ s ` : ‚ if s is a boundary edge, and v follows v as we traverse the corresponding boundarycomponent in a counterclockwise direction, r X s “ : Z s Z s ´ Z ´ s ` : ‚ if s is the internal edge of a self-folded triangle with the outer edge s , we set r X s “ r X s Given a puncture p and an edge s in ∆ ‚ , we say that s is adjacent to the notch at p if an edge-end of s is either the first or the last one at v i . We say that s is opposing thenotch at p if there exist edges s and s , both adjacent to notch at p , such that t s, s , s u form a triangle in ∆ ‚ , possibly self-folded or with some of the vertices coinciding. We then set (cid:15) p p s q “ $’&’% , if s is adjacent to the notch at p, ´ , if s is opposing the notch at p, , otherwise . If s is not the internal edge of a self-folded triangle, we now define α s “ ź p P P α (cid:15) p p s q p , whereas if s is the internal edge of a self-folded triangle with the internal puncture p and theouter edge s , we set α s “ α s α p . Now, let ∆ be the ideal triangulation underlying ∆ ‚ . Denote by X q ∆ the quantum chart ofthe cluster Poisson variety defined by ∆, see [GS19], and set X s to be the quantum cluster X -variable labelled by the edge s . We have: Theorem 4.21.
We have a well-defined homomorphism, ι ∆ ‚ : X q ∆ ÝÑ ζ SL p ∆ ‚ q ,X s ÞÝÑ : r X s α s : . Moreover, ι ∆ ‚ defines an isomorphism X q ∆ – χ PGL p ∆ q . Proof.
The map ι ∆ ‚ is a minor modification of the (quantum) cluster ensemble map, andcan be shown to be well-defined by a straightforward calculation with the powers in the q -commutation relations. It is also straightforward to see that ι ∆ ‚ p X q ∆ q Ă χ PGL p ∆ q .Let us now show that ι ∆ ‚ is surjective onto χ PGL p ∆ q . In what follows we will abusenotation and denote ι ∆ ‚ p X s q by X s . Given a puncture p and a triangle ∆ in ∆ ‚ , we set υ p p ∆ q “ , if two sides of ∆ are adjacent to the notch at p, , otherwise . Then, for a triangle ∆ with the sides s , s , s (two of which coincide if ∆ is self-folded) we set M ∆ “ : Z s Z s Z s ź p P P α υ p p ∆ q p : . From Corollary 4.8 and Proposition 4.20, we see that χ PGL p ∆ q is generated by α ˘ p for allpunctures p , and the products ź s Z n s s ź ∆ M n ∆ ∆ , n s , n ∆ P Z , which have zero weight with respect to the torus action at each puncture. At the same time,for any pair of triangles ∆ , ∆ which share a common edge s , we have M ∆ “ X s M ∆ ¨ ź s Z n s s ź p α n p p , for some n s , n p P Z . Therefore, for any fixed triangle ∆ in ∆, the quantum torus χ PGL p ∆ q is generated by X s for all edges s , α ˘ p for all punctures, and the products M n ∆ ∆ ź s Z n s s , n s , n ∆ P Z , invariant under the T -action at punctures. Note however, that the T -invariance conditionforces n ∆ to be even, and M itself is a product of elements of the form Z s and α p .Now, let γ “ p s , . . . , s n q be an even path in ∆, that is an ordered collection of edges,such that s i and s i ` share a common vertex v i for any i P Z { n Z . Define a product M γ “ n ź i “ Z p´ q i s i . Then the above discussion yields that χ PGL p ∆ q is generated by X s for all edges s , α ˘ p forall punctures p , and monomials M ˘ γ for all even paths γ . If p is not the internal puncture ofa self-folded triangle, one can check that α p “ : ź p PB s X s : , UANTUM DECORATED CHARACTER STACKS 51 where the product is taken over all edges s incident to p . Otherwise, if p is the internalpuncture of the self-folded triangle with the inner edge s and the outer one s , we get α p “ X s X ´ s . Similarly, we have M γ “ n ź i “ ź e i ă e ă e i ` X p´ q i e , where the product is taken over all edges e incident to the vertex v i and sitting between e i and e i ` with respect to the clockwise cyclic order at v i . This completes the proof of thesurjectivity of ι ∆ ‚ .Finally, let us prove that ι ∆ ‚ is injective. We note first that as a monomial map betweenquantum tori, the kernel is flat in q . Hence it suffices to show at q “ p χ PGL p ∆ qq isan algebraic torus of dimension | E | , where | E | is the number of edges in ∆, and is the dimensionof Spec p X q ∆ q . Indeed, we know that Spec p ζ SL p ∆ ‚ qq has dimension | P | ` | E | , where | P | is thenumber of punctures. Since the action of T on Spec p ζ SL p ∆ ‚ qq is free at every puncture, thedimension of Spec p χ SL p ∆ qq equals | E | , and so does the dimension of Spec p χ PGL p ∆ qq . (cid:3) The flip on Z p D q . The study of flips between triangulations of S starts with thequadrilateral D and its two triangulations ∆ and ∆ . An arbitrary flip can be understood interms of the one between ∆ , ∆ , using excision to isolate the pair of adjacent triangles wherethe flip takes place. In what follows, we retain our convention from Section 3.8 of enumeratingthe T –regions t , , , u of D in counter-clockwise order.Recall from Proposition 3.51 that the category Z p D q is equivalent to the orthogonalcomplement in F q p Conf fr q´ mod G ˆ T of the localizing subcategory of torsion modules. ByTheorem 3.62, we have full reflective subcategories Z p ∆ q , Z p ∆ q of Z p D q , each equivalentto the full subcategory of F q p Conf fr q –modules on which all elements t ∆ e u associated to theedges of the corresponding triangulation act invertibly. By the same argument used to proveTheorem 3.62, there is another open reflective subcategory Z p ∆ , ∆ q Ă Z p D q ã Ñ F q p Conf fr q´ mod G ˆ T , equivalent to the full subcategory of F q p Conf fr q –modules on which all elements D ij with1 ď i ă j ď Z p ∆ , ∆ q we open four gates in each T -region, therebyendowing D with a T -action. We denote by Dist ∆ , ∆ the restriction of the distinguishedobject, and we set r ζ p ∆ , ∆ q “ End T p Dist ∆ , ∆ q , giving an equivalence, Z p ∆ , ∆ q » r ζ p ∆ , ∆ q´ mod T and the following: Proposition 4.22.
The transition functor between charts Z p ∆ q and Z p ∆ q is given by: µ ∆ , ∆ : Z p ∆ q ÝÑ Z p ∆ q m ÞÝÑ ζ p ∆ ‚ , ∆ q b ζ p ∆ ‚ q m. This transition functor can be described completely explicitly as follows. Consider theideal tetrahedron whose four triangular faces are partitioned into two copies of D . On theboundary of this tetrahedron we have a fencing graph Γ ∆ , ∆ illustrated in Fig. 10. a p q a p q a p q a p q a p q a p q a p q a p q a p q a p q a p q a p q A A A A A A A A Figure 10.
The fencing graph Γ ∆ , ∆ in Proposition 4.23 Proposition 4.23.
The algebra r ζ p ∆ , ∆ q has generators A ˘ (cid:96) with (cid:96) P E l p Γ ∆ , ∆ q and a ˘ e with e P E s p Γ ∆ , ∆ q , in correspondence with the edges of the fencing graph Γ ∆ , ∆ on the surfaceof the ideal tetrahedron shown in Fig. 10. These generators satisfy the fencing graph relationsdescribed in Proposition 4.11, and in addition to these the exchange relation Z , Z , “ q ´ Z , Z , ` q Z , Z , , (4.5) where Z , “ : a p q a p q a p q A , : , Z , “ : a p q a p q a p q A , : , Z , “ : a p q A , a p q a p q : ,Z , “ : a p q a p q a p q A , : , Z , “ : a p q a p q a p q A , : , Z , “ : a p q a p q A , a p q : . In particular, we have injective algebra homomorphisms r ζ p ∆ q Ñ r ζ p ∆ , ∆ q , r ζ p ∆ q Ñ r ζ p ∆ , ∆ q , corresponding at level of generators to the inclusions of fencing graphs Γ ∆ Ă Γ ∆ , ∆ Ą Γ ∆ . Proof.
The generators of r ζ p ∆ , ∆ q are defined in complete analogy with Propositions 4.6and 4.7. For instance, if we enumerate the tensor factors in the i –th T -region of D in clockwiseorder, then for boundary long edges ij we have A i,j “ D i,j b χ p i qp , , , q b χ p j qp , , , q , while a p i q j “ χ p i q ε j ´ ε j ` . The diagonal long edge variables are A , “ D , b χ p qp , , , q b χ p qp , , , q , A , “ D , b χ p qp , , , q b χ p qp , , , q ,A , “ D , b χ p qp , , , q b χ p qp , , , q , A , “ D , b χ p qp , , , q b χ p qp , , , q . The fencing graph relations easily follow from this description, the q -commutation relationsamong the D i,j in Corollary (3.56), and the multiplication rule (3.5). The exchange relationfollows by observing that Z i,j “ q ´ D i,j b χ p i qp , , , q b χ p j qp , , , q , UANTUM DECORATED CHARACTER STACKS 53 and then applying the three-term relation in F q p Conf fr q D , D , “ q ´ D , D , ` q D , D , established in Corollary 3.57. (cid:3) Remark 4.24.
The exchange relation (4.5) can be rewritten in terms of Weyl orderedquantum torus elements as Z , “ : Z ´ , Z , Z , : ` : Z ´ , Z , Z , : , (4.6)or alternatively as Z , “ : Z ´ , Z , Z , : ` : Z ´ , Z , Z , : . Flips on Z p S q . Now let us consider a general decorated surface S , and two notchedtriangulations ∆ ‚ and ∆ differing by the flip of a single edge. Let us denote by r S thedecorated surface obtained as union of all but the two triangles sharing the edge, and by r ∆the resulting triangulation of r S . Then we have a decomposition S “ r S \ D \ D . By excision, the flip functor µ ∆ , ∆ thus admits a natural description as µ ∆ , ∆ “ id r S b µ D , where µ D is the functor between open subcategories of Z p D q from Proposition 4.22 inducedby changing its triangulation. The only subtlety arises when we wish to describe this functorat the level of generators and relations for the algebras ζ p ∆ ‚ q , ζ p ∆ q . Recall that if S haspunctures, it is the the data of the notching that singles out a distinguished set of generatorsfor the quantum torus ζ p ∆ ‚ q . Keeping track of the effect of flips on the notch necessitates asmall amount of additional bookkeeping, which we shall now describe.Let us again enumerate the (possibly coincident) T -regions of the 4-gon t , , , u incounterclockwise order. Then we have edges t e , , e , , e , , e , u common to both triangulations∆ ‚ , ∆ , where B e i,j “ t i, j u and we allow for the possibility that some of the e i,j are in factidentified. Using the notchings, we assign to each edge–end e i of a diagonal of D an integer γ i defined by γ i “ $’&’% | e i | is minimal, ´ | e i | is maximal , . (4.7)More informally, we declare γ i “ e i is the first edge-end with respect to the total orderfrom the notched triangulation in which e i appears, while γ i “ ´ e i is the last edge-endwith respect to the notching; in all other cases we set γ i “
0. Then we have the followingexchange relation between the distinguished generating sets of the quantum tori ζ p ∆ ‚ q , ζ p ∆ q corresponding to the notched triangulations ∆ ‚ , ∆ : Proposition 4.25.
Suppose that ∆ ‚ , ∆ are two notched triangulations whose underlyingtriangulations differ by flipping a single edge. Let Z i,j P ζ p ∆ ‚ , ∆ q denote the elementcorresponding to the diagonal e i,j with edge-ends e i and e j , and let p γ i q i “ be the integersdefined in (4.7) . Let us set α “ α δ γ , ´ α ´ δ γ , α δ γ , ´ α ´ δ γ , ,α “ α ´ δ γ , α δ γ , ´ α ´ δ γ , α δ γ , ´ . Then we have the following exchange relations in ζ p ∆ ‚ , ∆ q : Z , “ : α Z ´ , Z , Z , : ` : α Z ´ , Z , Z , : , (4.8) Z , “ : α Z ´ , Z , Z , : ` : α Z ´ , Z , Z , : , (4.9) where the Weyl ordering is taken with respect to the set of generators t Z e , α i u . Proof.
We must translate the identity (4.6), which holds in the larger algebra r ζ p ∆ , ∆ q » r ζ p ∆ q b ζ p D q b r ζ D p ∆ , ∆ q , into an identity between elements of the subalgebra ζ p ∆ ‚ , ∆ q of T ∆ –invariants in r ζ p ∆ , ∆ q .To do this, for each T –region i of D , let w i be the first vertex of the fencing graph for D with respect to the clockwise order within i . Moreover, let v i be the distinguished vertex inthe T -region of S that contains the T –region i of the subsurface D . Now write π p i q for theset of short edges that make up the minimal path in the fencing graph Γ ∆ , ∆ starting at v i and travelling clockwise to vertex w i . Then multiplying both sides of the relation (4.6) by theproduct of short edges : ź e P π p q ź f P π p q a e a f : , we obtain the relation (4.8). (cid:3) The following is an easy direct computation.
Corollary 4.26.
The isomorphisms ι ∆ ‚ between X q ∆ and χ PGL p ∆ q intertwine the clustermutation and the restricted flips functors χ PGL p ∆ , ∆ q obtained from ζ SL p ∆ ‚ , ∆ q . Topological remarks.
In this final section we collect a number of remarks concerningthe topological invariance built into our construction.Let Aut p S q denote the 2-group of automorphisms of S in S urf, i.e. the category with a singleobject, whose 1-morphisms are self-diffeomorphisms of S which respect the stratification andthe labeling by G and T , and whose 2-morphisms are stratified isotopies of such (equivalentlyAut p S q may be regarded as a monoidal category in which all objects and morphisms areinvertible). We note that the 1-truncation π ď Aut p S q is the precisely the marked mappingclass group, since by definition this means that we take isotopy equivalence classes. Givena set G of G - and T -gates, let us also denote by Aut G p S q the sub 2-group of Aut p S q ofdiffeomorphisms and isotopies which fix G componentwise.Simply because Z is a functor, the category Z p S q inherits a canonical action of the 2-groupAut p S q . This data can be understood as an action of the mapping class group, in whichfunctors compose by the group law only up to a coherent isomorphism, encoded by a 2-cocyclewith values in the center of the category. We note that the distinguished object, being definedby the empty embedding is preserved canonically (i.e. up to a canonical isomorphism) byany diffeomorphism, so that it obtains a canonical structure of a Aut p S q -fixed point of thecategory. In particular the mapping class group acts strictly on the vector space End p Dist S q ,and in such a way that it intertwines the cluster charts coming from different triangulations. UANTUM DECORATED CHARACTER STACKS 55
Suppose we are given two triangulations ∆ and ∆ related by an isotopy γ . We may assume γ at time t “ S , so that as t runs to t “
1, it inducesisomorphisms between objects of Z p ∆ q and Z p ∆ q . Since both categories are closed underisomorphisms, it means two subcategories related by an isotopy are simply equal.An interesting extra feature comes from the Rep q p G q -action at the gates. In the presenceof gates G , any functor arising coming from Aut G p S q is canonically promoted to a G -modulefunctor, and any isotopy to a G -module natural isomorphism. This additional naturality hasa consequence for the charts ζ p ∆ ‚ q and their mutations. The algebras ζ p ∆ ‚ q are defined withrespect to a notched triangulation, yet their presentation, as encoded by the fencing graph,depended only on the associated notched ideal triangulation. We note that two isotopy classesof notched triangulations give rise to the same notched ideal triangulation if and only if theydiffer by iterated application of the Dehn twist implementing 2 π rotation at some puncture.This Dehn twist preserves the notching, hence it gives a Rep q p T q -module auto-equivalence of Z p S q . However, the natural isotopy undoing the Dehn twist does not preserve the notch.This means that the Dehn twist is isomorphic to the identity functor as a plain functorof categories, but is not isomorphic as a Rep q p T q -module category. Hence the Dehn twistdoes not act trivially on internal endomorphism algebras such as ζ p ∆ ‚ q . Indeed, a directcomputation implies that the Dehn twist induces a homomorphism r ζ p ∆ ‚ q Ñ r ζ p ∆ q given by Z e ÞÑ qZ e α p , for all edges e incident to the puncture.And additional interesting subtlety comes when we consider the appearance of topologicalframings of S . In Section 4.3, we have implicitly fixed a 2-framing of the surface S whencomputing the charts r ζ . The data of the 2-framing can be prescribed by a jellyfish diagramas in Fig. 9: the triangles are drawn in some order in a line, and the attaching digons arestretched out to long legs connecting the triangles. Such a prescription topologically fixes a2-framing on the surface – the blackboard framing – and also a total ordering on edge-ends ofthe triangulation. This data entered into our computation of the generators and relations for ζ p ∆ ‚ q . However, the reader will note that the algebra r ζ p ∆ q has been defined in such a waythat it does not depend on this data of framing.Hence, while neither the algebras r ζ p ∆ q , ζ p ∆ ‚ q , nor the flips r ζ p ∆ , ∆ q , ζ p ∆ ‚ , ∆ q dependon the choice of 2-framing, their presentation by generators and relations (that is, theirdistinguished choice of generators), do indeed depend on such a choice (in fact, less data, thechoice of a 3-framing at the quantum level, a spin structure classically). Algebraically, this isbecause we have chosen the elements D ij as generators of a 1-dimensional space of invariantsin F q p N z G q b F q p N z G q , but this space doesn’t naturally come with a distinguished vector.We wish to stress that in traditional approach to quantization via quantum cluster algebras,one must fix these choices a priori in order to define the quantum cluster charts and their flipsin the first place. By contrast, we may make such choices in order to compute our quantumcluster charts and their flips a posteriori .This distinction also resolves a potential point of confusion about A -varieties in our set-up.Traditionally, these are defined as moduli spaces of twisted local systems on the surface with B -reductions. Any choice of spin structure on the surface (in particular any 3-framing, and inparticular any 2-framing) gives an isomorphism between the moduli spaces of twisted andordinary local systems. If defining charts and flips by formulas, one must work with twistedlocal systems and non-canonically identify them with ordinary local systems by a choice ofspin structure. In our approach, this is reversed, and ordinary local systems are what naturallyappear. α α α α Figure 11.
At left: a cylinder with a marked point on each boundary circle.In the middle: a punctured torus. At right: a 3-punctured sphere.5.
Examples
In this section we consider the examples when S is a cylinder with one marked point oneach boundary circle, a punctured torus, and a three punctured sphere. On Fig. 11, we showa pair ∆ , ∆ of superimposed triangulations of each surface. The triangulation ∆ consists ofedges labelled t , , u or t , , , u , and ∆ is obtained from ∆ by flipping the edge 3 to theone labelled 3 . We denote the tori ζ SL p ∆ q and ζ SL p ∆ q by ζ and ζ respectively. Similarly,we let χ and χ denote the tori χ PGL p ∆ q and χ PGL p ∆ q . We let Z j , α p be the generatorsof ζ , while setting Z : “ Z P ζ . The generators of χ and χ are denoted respectively by t X i u and t X i u . Finally, we let µ : “ µ ∆ , ∆ be the flip between the triangulaitons ∆ and ∆ .5.1. The cylinder.
We have ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , Z ˘ D ,ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , Z ˘ D , where the elements Z and Z are central in both tori, the other relations read Z Z “ qZ Z and Z Z “ qZ Z , and the flip µ takes form Z Z “ Z Z ` qZ . The X -variables of the torus χ then read X “ : Z Z Z ´ : , X “ : Z Z ´ Z ´ : , X “ : Z Z Z ´ : , X “ : Z Z Z ´ : , and satisfy the relations X X “ q X X , X X “ q ´ X X , X X “ X X ,X X “ q X X , X X “ q ´ X X , X X “ q X X . Similarly, the X -variables of the torus χ are given by X “ : Z Z Z ´ : , X “ : Z Z Z ´ : , X “ : Z Z ´ Z ´ : , X “ : Z Z Z ´ : , and satisfy the relations X X “ q ´ X X , X X “ q X X , X X “ X X ,X X “ q ´ X X , X X “ q X X , X X “ q ´ X X . Finally the flip µ reads X “ X ´ , X “ X p ` qX q , X “ X p ` qX q , UANTUM DECORATED CHARACTER STACKS 57 and X “ X p ` qX ´ q ´ p ` q X ´ q ´ . The punctured torus.
In this case, we get ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , α ˘ D ,ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , α ˘ D . The generators satisfy relations Z Z “ qZ Z , Z Z “ qZ Z , Z Z “ qZ Z ,Z Z “ qZ Z , Z Z “ qZ Z , as well as αZ x “ q Z x α for any edge x P t , , , u . The flip µ reads Z Z “ q Z α ` qZ . The X -generators of the torus χ read X “ : Z Z ´ α : , X “ : Z Z ´ α : , X “ : Z Z ´ α ´ : , and satisfy the relations X i X i ` “ q X i ` X i , for i P Z { Z . Similarly, the X -generators of the torus χ take form X “ : Z Z ´ α : , X “ : Z Z ´ α ´ : , X “ : Z Z ´ α : , and satisfy the relations X i X i ` “ q ´ X i ` X i , for i P Z { Z . Finally, the flip of X -variables is X “ X p ` qX qp ` q X q , X “ X p ` qX ´ q ´ p ` q X ´ q ´ , X “ X ´ . The 3-punctured sphere.
We have ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , α ˘ , α ˘ , α ˘ D ,ζ “ C r q ˘ s @ Z ˘ , Z ˘ , Z ˘ , α ˘ , α ˘ , α ˘ D . The generators satisfy relations Z Z “ q Z Z , Z Z “ q Z Z , Z Z “ q Z Z ,Z Z “ Z Z , Z Z “ qZ Z , as well as r α i , α j s “ α i Z j “ q ´ δ ij Z j α i , for any 1 ď i, j ď
3. The flip µ reads Z Z “ q : α α Z Z : ` q ´ : α Z Z : . The torus χ is commutative and is generated by the elements X j “ α α α α j , for 1 ď i, j ď
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