#### Quantum character varieties and braided module categories

Quantum character varieties and braided module categories
David Ben-Zvi 0 1 2
Adrien Brochier 0 1 2
David Jordan 0 1 2
B David Jordan 0 1 2
0 School of Mathematics, University of Edinburgh , Edinburgh , UK
1 MPIM , Bonn , Germany
2 Department of Mathematics, University of Texas , Austin, TX 78712-0257 , USA
We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants S A of a surface S, determined by the choice of a braided tensor category A, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for A, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided A-modules are objects of the torus category T 2 A. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A = Repq G with the category Dq (G/G)-mod of equivariant quantum D-modules. When G = G L n , we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra SHq,t . Mathematics Subject Classification 17B37 · 16T99 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Braided module categories and quantum moment maps . . . . . . . . . . . . . . . . . . . . . .
Contents
1.2 Computing factorization homology of marked surfaces . . . . . . . . . . . . . . . . . . . . . .
1.3 Quantization of character varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Towards quantum character sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The double affine Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Factorization homology of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Factorization homology of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Marked points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Braided module categories and surfaces with marked points . . . . . . . . . . . . . . . . . . . . . .
3.1 The oriented case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Reconstruction theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Reconstruction from tensor functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Reconstruction for the annulus category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Braided module structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Closed surfaces, markings and quantum Hamiltonian reduction . . . . . . . . . . . . . . . . . . . .
5.1 Marked points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The functor of global sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Let S denote a topological surface and G a reductive group. The G-character stack
ChG (S) of S is the moduli space of G-local systems on S, the quotient of the affine
scheme of representations of the fundamental group of S into G by the conjugation
action of G. Character stacks—and their variants associated to surfaces with marked
points or other decorations, which we collectively refer to as character varieties—play
a central role in geometry, representation theory and physics. A crucial feature of
character stacks is their local nature—they are obtained from gluing stacks of local
systems on patches of S. As a result they provide a natural source of topological
field theories: numbers, such point counts/Euler characteristics of character varieties
appear in two-dimensional field theory; vector spaces such as sections of line bundles
on character varieties appear in three-dimensional field theory; and finally, categories
of sheaves on character varieties appear naturally in four-dimensional field theory.
We will be concerned with the four-dimensional setting, accessing character
varieties through their categories of coherent sheaves, as they appear in the Betti form of
the Geometric Langlands program and in twisted d = 4 N = 4 super-Yang-Mills
theory, following the work of Kapustin-Witten [
13
]. Our goal is to construct and describe
quantum character varieties—q-deformations of these categories, which quantize the
Goldman symplectic structure on the character stack (associated to a choice of
invariant form on g). Moreover we endow these quantum character varieties with all of
the structures expected from their origin in topological field theory, and develop their
study as a natural setting for a variety of constructions in quantum algebra.
In [
10
], we initiated the construction of quantum character varieties via factorization
homology of braided (or balanced) tensor categories A on topological surfaces: this
produces category-valued invariants of framed (or oriented) topological surfaces, with
the desired strong functoriality and locality properties. Starting from the braided tensor
category A = Repq G of integrable representations of the corresponding quantum
group yields the desired functorial quantizations. We computed these invariants for
unmarked punctured surfaces, in terms of certain explicitly presented, and in many
cases well-known, quantum algebras, which were constructed as certain twisted tensor
products of the so-called “reflection equation algebra” FA ∈ A.
In the present paper, we extend this framework to the setting of closed surfaces, as
well as to surfaces with marked points. In brief, our main results are as follows:
• The possible markings of points (codimension two defects) in the topological field
theory defined by A are given by module categories over the monoidal category
Ann A. In [
10
] we identified the underlying category with modules FA -modA for
the reflection equation algebra of A. In Sect. 4.2 we explicitly identify the new
induced monoidal structure on this category, the field-goal tensor product.
• We show in Theorem 3.11 that codimension two defects ( Ann A-modules) are
identified with braided module categories over A (in the sense of [
7,8,23
]), in
the same way that the unmarked disc is assigned A (see Sect. 2.1 for an
introduction to factorization homology of marked surfaces). There are many natural
examples of braided module categories (see below), including ones
corresponding to versions of character varieties with parabolic structures, fixed conjugacy
classes, or other boundary conditions (codimension one defects wrapping a
circle). They play the role for the 4d Kapustin-Witten (Betti Geometric Langlands)
TFT that integrable representations of the loop group play for the 3d Chern-Simons
(Witten-Reshetikhin-Turaev) theory.
• In Theorem 1.1, we identify braided module categories with a generator as modules
for an algebra object AM ∈ A equipped with a “quantum moment map”, i.e. an
algebra homomorphism, μ : FA → AM. As we explain, μ is a quantum version
of a group-valued moment map appearing in the classical setting [
3
].
• We describe (in Theorems 4.1, Proposition 4.3, and Corollary 4.8) the result of
gluing braided module categories with generators over their common braided
Aaction as a category of bimodules in Ann A. The quantum moment maps play
a key role in defining the bimodule structure, and the resulting categories may
be regarded as categorical quantum Hamiltonian reductions, along the respective
quantum moment maps.
• In particular, we compute the “global functions” on general quantum character
varieties: the endomorphisms of the quantum structure sheaf on a closed (or marked)
surface are identified as the quantum Hamiltonian reduction of the algebra AS◦
associated to a punctured surface along the corresponding quantum moment map.
• The torus integral T 2 A of a balanced tensor category A is identified with the
trace (or Hochschild homology) of the 2-category of braided A-modules, i.e., the
natural receptacle for characters of braided modules. For A = Repq G we identify
the torus integral with the category Dq (G/G) -mod of adjoint-equivariant
quantum D-modules, thus providing an interpretation for these characters as quantum
analogues of character sheaves.
• For G = G Ln, the category Dq (G/G) -mod has a “mirabolic” version obtained by
marking a single point in T 2 by the quantum “Ruijsenaars–Schneider” conjugacy
class N = At -modA. We show that we recover Cherednik’s spherical double
affine Hecke algebra SHq,t , as the “global functions” (endomorphisms of the
quantum structure sheaf). Hence, the global sections of any mirabolic quantum
D-module carries a canonical action of the spherical double affine Hecke algebra.
1.1 Braided module categories and quantum moment maps
Factorization homology provides a general mechanism to construct invariants of
nmanifolds starting from algebras A over the En (little n-disks) operad—i.e., objects
which carry operations labeled by inclusions of disks into a larger disk (see Sect. 2.1
for a brief review). The invariant M A of an n-manifold is then defined as the universal
recipient of maps from A for every inclusion of a disk into M , and factoring through
the operad structure.
There is a natural operadic notion of module M for an En-algebra A, captured
pictorially by placing the module at a marked point of a disk and allowing insertions
of A at disjoint disks. It is well-known (see e.g. [
2,35
]) that the structure of
Enmodule over an En-algebra A on M is equivalent to the structure of left module over
the associative (E1) “universal enveloping algebra” U (A), namely the factorization
homology U (A) = Ann A of an annulus with coefficients in A. The latter category
is equipped with an E1 (monoidal) structure, coming from concatenation of annuli.
This is the structure used in the excision axiom on Ann = Sn−1 × I (see Sect. 3 for
more details) by which one computes factorization homology.
In [
10
] we initiated the study of factorization homology of surfaces with coefficients
in braided tensor categories, which are precisely the E2-algebras in a certain 2-category
C = Pr of linear categories with the Kelly-Deligne tensor product. (More precisely,
braided tensor categories A can be integrated over framed surfaces, while equipping
A with a balanced structure extends this integral to oriented surfaces.) In the same
way, factorization homology of surfaces with marked points demands that for each
marked point we give an E2-module M over our chosen braided tensor category A.
In Theorem 3.11, we show that in the case C = Pr, the notion of an E2-module is
equivalent to that of a “braided module category”, a concept introduced in [
7,8,23
],
and closely related to the reflection equation algebra. We also introduce the notion of a
“balanced braided module category”, which captures the structure of a Disk2or -module
(i.e. the oriented marked case), and we show in Theorem 3.12 that, when A itself is
balanced, it endows any of its braided module categories with a canonical (though not
unique) balancing.
Examples of braided module categories include the following:
(1) The category A itself is a braided A-module. It corresponds to the “vacuum
marking”, and is an essential component in our computation for unmarked surfaces.
(2) For any surface S◦ with circle boundary, the category S◦ A is a braided module
category, by insertions of annuli along the boundary. In [
10
], we identified S◦ A
with the category of modules for an explicitly constructed algebra AS◦ .
(3) Quantizations of conjugacy classes in G, following [
18–20,39
], define braided
module categories. An important example is the so-called Ruijsenaars–Schneider
conjugacy class, consisting of matrices which differ from the identity by a matrix
of rank at most one [
6,38,46
].
(4) Examples of braided module categories related to a variant of the trigonometric
Knizhnik–Zamolodchikov equation, and the theory of dynamical quantum groups,
appear in [
7,8,23
].
(5) A “boundary condition” is the local marking data for a half-plane. Algebraically,
this is the data of a tensor category B attached to the boundary, together with
a braided tensor functor A → Z (B) to the Drinfeld center of B. The trace,
or Hochschild homology category, of such a B carries the structure of a braided
module category for A. Important examples are provided by parabolic subgroups.
In [
10
], we identified Ann A with the category of modules for the “reflection
equation algebra” FA ∈ A. In the case A = Repq G, FA is a quantization of the
coordinate algebra O(G), equipped with its Semenov-Tian-Shansky Poisson bracket.
In Sect. 4.2, we prove the following theorem, giving yet a third reformulation of the
notion of a braided module category, in terms of FA:
Theorem 1.1 Let M be a braided module category. (1) For every M ∈ M, we have a canonical homomorphism of algebras,
μM : FA → EndA(M ).
We call μM the quantum moment map attached to M .
(2) Assuming M is a progenerator for the A-action1, we moreover have an
equivalence of Ann A-module categories,
(3) The action of any X ∈ Ann A on any N ∈ M
by relative tensor product,
EndA(M ) -mod
Ann A
is given
M
EndA(M ) -mod
Ann A
.
N
X → N ⊗ X ,
FA
over the homomorphism μM .
Conversely, given an algebra A ∈ A and a homomorphism μ : FA → A, the category
M = A -modA is equipped with the structure of a braided module category, with
action as in (3). The regular A-module A ∈ M is an A-progenerator in this case.
Natural examples of quantum moment maps arise in the following contexts:
(1) The quantum moment map for A itself is the co-unit homomorphism, : FA →
1A, which quantizes the homomorphism of evaluation of a function at the identity
element.
(2) In Sect. 4, we obtain canonical quantum moment maps μ : FA → AS◦ which
control the braided module category structure on S◦ A. These quantize the
classical multiplicative moment maps, which send a local system to its monodromy
around the puncture.
1 Equivalently, for the Ann A-action; see Theorem 4.3.
(3) By their construction—as equivariant quotients of the reflection equation
algebra—quantizations of conjugacy classes carry canonical quantum moment
maps.
1.2 Computing factorization homology of marked surfaces
Let us fix a surface S equipped with a marked point x ∈ S, a disc Dx containing x , and
a braided module category M. Let S◦ = S\x , and fix a disc embedding ix : Dx ⊂ S,
and resulting annulus embedding Ann Dx \x ⊂ S. We may then compute the
factorization homology using excision,
(A, Mx )
(S,x)
A
S◦
A
Ann
Mx .
Building on Theorem 1.1 we can describe the tensor product above explicitly, in
the language of quantum Hamiltonian reduction.
Theorem 1.2 Let ix∗ : M → (S,x)(A, M) denote the push-forward in factorization
homology along the embedding ix .
(1) For any M ∈ M, we have a natural isomorphism:
End (ix∗(M )) ∼= AS◦
μ
M := Hom(1A, AS◦ F⊗A EndA(M )),
of the endomorphism algebra of ix∗(M ), in the category (S,x)(A, Mx ) with the
quantum Hamiltonian reduction of AS◦ along the quantum moment map μM .
(2) Suppose that M is an A-progenerator. Then we have equivalences of categories,
(A, M)
(S,x)
A
S◦
A
Ann
M
( AS◦ -mod- End(M ))FA -mod,
with the category of bimodules for AS◦ and End(M ), in the category FA -mod.
(3) The quantum global sections functor
= Hom(i∗(M ), −) :
S
A → ( AS◦
μ
M )op -mod,
valued in the category of modules for the quantum Hamiltonian reduction is
naturally equivariant for actions of the marked-and-colored mapping class group
of the surface.
This Theorem (more generally in the case of several marked points) is proved in
Sect. 4.
In particular we obtain a description of the category associated to a closed unmarked
surface S. Choose some disk D2 ⊂ S, and let S◦ denote its complement in S.
Corollary 1.3 We have an equivalence of categories,
S
A
A
S◦
A
Ann
A
( AS\D2 -mod- 1A)FA -mod,
with the category of bimodules for AS◦ and AD2 = 1A, in the category FA -mod.
Likewise we can identify global functions on the quantum character variety (i.e.,
endomorphisms of the quantum structure sheaf OA,S ) with the quantum Hamiltonian
reduction of AS◦ along the quantum moment map μ,
End(OA,S ) =∼
AS◦
AS◦ · μ(ker( ))
inv
.
Thus we have a global sections functor
= Hom(OA,S, −) :
S
A → (AS◦
1 )op -mod,
μ A
valued in the category of modules for the quantum Hamiltonian reduction, equivariant
for an action of the mapping class group of the surface.
1.3 Quantization of character varieties
In the classical setting A = Rep G, it was a fundamental observation of [
3
] that G
character varieties of closed surfaces could be obtained via “multiplicative
Hamiltonian reduction” of their punctured counterparts. Let us briefly recall the classical
construction here.
Let S◦ be a surface with one distinguished circle boundary component with a point
p chosen on it. Let RG (S◦) denote the representation variety of S◦, i.e.
RG (S◦) = {ρ : π1(S, p) → G}.
Equivalently, RG (S◦) is the variety of G-local systems on S◦, equipped with a
trivialization of the fiber at p. Changing the choice of trivialization amounts to conjugating
a given homomorphism by a group element. Hence, the G-character stack of S is the
quotient stack
ChG (S◦) = RG (S◦)/G.
μ : RG (S◦) −→ RG ( Ann) = G.
The embedding of the annulus around the circle boundary of S induces a G-equivariant
map,
The map μ is called a “multiplicative” or “group-valued” moment map, in [
3
].
Fix a conjugation invariant subvariety C ⊂ G (i.e., a union of conjugacy classes).
Then μ−1(C ) ⊂ RG (S) is a G-stable subvariety. The character stack of (S, C ) is then
the quotient stack
ChG (S, C ) = μ−1(C )/G.
In other words, ChG (S, C ) is a moduli stack of G-local systems on S◦ whose
monodromy around the boundary lies in C . By definition, the category QCoh(ChG (S, C ))
is the category of G-equivariant quasi-coherent sheaves on μ−1(C ). The variety
μ−1(C )/G is called the “multiplicative Hamiltonian reduction” of RG (S◦) along
μ.
The main results of [
10,12
], give identifications,
S◦
G -mod
QCoh(Ch(S◦)),
AS◦ =∼ O(G2g).
The category QCoh(C /G) is a braided module category for Rep G, which we may
associate to the puncture, and the category QCoh(μ−1(C )/G) is precisely the category
produced by factorization homology for a marked surface (S, x ), where the marked
point is decorated with QCoh(C /G).
When we instead take A = Repq G, excision gives rise to “quantum multiplicative
Hamiltonian reduction”, as discussed in e.g. [
38
]. Therefore, we obtain quantizations
of character stacks of closed surfaces. Taking global sections passes to the
affinization of the character stack, the Poisson variety obtained as the categorical quotient
RG (S) // G. More generally if C is any conjugacy class, the categorical quotient
μ−1(C ) // G is a symplectic leaf of the Poisson variety RG (S) // G (the case of the
closed surface corresponds to C = {e}. In [
18–20
] an explicit quantization of any
given conjugacy class is given, using Verma modules: by construction these come
equipped with an equivariant algebra map from Oq (G). By Theorem 1.1 the category
of equivariant modules over this algebra is a braided module category, hence its
factorization homology over the marked surface as above gives a quantization of the variety
μ−1(C ) // G.
1.4 Towards quantum character sheaves
The invariant assigned to the torus S = T 2 plays a central role in topological field
theory. In three-dimensional Chern-Simons/Witten-Reshetikhin-Turaev theory at level
k the invariant of T 2 is identified on the one hand with the Verlinde algebra, the group
of characters of integrable level k representations of the loop group (which themselves
form the invariant of S1, i.e., the Wilson lines or codimension two defects). The natural
mapping class group symmetry of the torus invariant then explains the well-known
modularity of these characters. On the other hand the Freed–Hopkins Teleman theorem
[
26
] identifies the T 2 invariant with a version of class functions on the compact group,
namely the twisted equivariant K -theory of Gc/Gc. In this section we describe the
corresponding roles of the quantum character variety of the torus.
Let us consider the oriented field theory defined by integrating a balanced tensor
category A on oriented surfaces. We have identified codimension two defects for
quantum A-character varieties with braided A-modules, i.e., modules for U (A) =
Ann A, which thanks to the balancing is also identified (as monoidal category) with
the cylinder integral Cyl A (see Remark 3.6). In the language of extended topological
field theory, this means we attach the 2-category U (A) -mod to the circle.
The excision axiom applied to a decomposition of T 2 into cylinders allows us
to identify the torus integral as the monoidal Hochschild homology, or trace, of the
cylinder (hence annulus) integral of A:
T 2 A
U (A)
U (A) U (A)op U (A) = T r (U (A)).
Equivalently, we can describe the torus integral as the trace (or Hochschild
homology) of the 2-category of braided A-modules. It follows from the general theory of
characters in Hochschild homology (see e.g. [
12
] for references) that T 2 A carries
characters [M] ∈ T 2 A for sufficiently finite (i.e., dualizable) braided A-modules
M.
Remark 1.4 (Braided G-categories and loop group categories) The 2-category of
braided Repq G-modules is a 4d gauge theory analog of the category of integrable
level k representations of the loop group. Indeed, it is the Betti form [
13
] of the
2category of chiral module categories over the Kazhdan-Lusztig category of integrable
representations of the loop algebra, which itself is a form of the local geometric
Langlands 2-category of categories with an action of the loop group at level k (where q is
essentially exp(2π i /k)), see [
33
].
In the case A = Repq G, in [
10
] we identified the punctured torus category with
the category of modules in Repq G for the algebra of quantum differential operators,
considered as an algebra object in Repq G under the adjoint action:
T 2\D2
Repq (G)
Dq (G) -modRepq G .
Note that since we are considering modules in Repq G rather than V ect , this category
is a quantum analog not of the category of D-modules on G but of the category of
D-modules on G which are weakly equivariant for the adjoint action (from which
the former can be obtained by de-equivariantization). It follows from the quantum
Hamiltonian reduction formalism of the previous section that sealing up the puncture
results in imposing the quantum moment map relations for the adjoint action—i.e., in
imposing strong equivariance.
We define:
Definition 1.5 The category Dq ( GG ) -mod of strongly ad-equivariant Dq (G)-modules
has its objects pairs (M , φ) consisting of a Dq (G)-module M ∈ Dq (G) -modA, and
an action map, φ : M O⊗A 1A → M in the category Dq (G) -modA, satisfying the
associativity conditions making M into an 1A-module.
As a corollary of Theorem 1.1, we have:
Theorem 1.6 We have equivalences of categories,
T 2
Repq G
T r (U (A))
Dq
G
G
-mod .
In particular Dq ( GG ) -mod inherits an action of S L2(Z), including a quantum Fourier
transform (S-transformation) generalizing the difference Fourier transform in the case
G = H a torus. Indeed the endomorphisms of the quantum structure sheaf are known
in many cases (see below for the t -analog) to recover
End(ORepq G,T 2 ) ∼= Dq (H )W ,
the algebra of W -invariant q-difference operators on the torus H . This should be
compared to the computation of [
24
].
It follows that, in analogy with the Freed–Hopkins–Teleman theorem for Chern–
Simons theory, the characters of braided Repq G-modules form quantum D-modules
on G/G. This is a quantum analog of the interpretation [
11
] of Lusztig character
sheaves in D(G/G) -mod in terms of module categories for D-modules on G. We
likewise expect a theory of quantum character sheaves to provide a natural q-analog
of the Lusztig theory. Interesting examples of such quantum character sheaves are
provided by the quantum Springer sheaves—the characters of the braided module
category tr (Repq B) associated to the Repq G-algebra defined by the quantum Borel
(or other parabolics), which can be expected via a quantum Hotta-Kashiwara theorem
to be described by a q-analog of the Harish-Chandra system. In particular one expects
G ) -mod to carry a “quantum generalized Springer”
the entire torus category Dq ( G
orthogonal decomposition into blocks labeled by cuspidal objects associated to Levi
subgroups, in analogy with the results of [
37
] for D(g/G).
1.5 The double affine Hecke algebra
The double affine Hecke algebra (abbreviated DAHA, and denoted Hq,t ) associated to
G = G L N (or more generally to a reductive group G) is a celebrated two-parameter
deformation of the group algebra of the double affine Weyl group of G, introduced by
Cherednik. It contains as a subalgebra the spherical DAHA (denoted SHq,t ), which is
a flat one-parameter deformation of the algebra Dq (H )W of W -invariant q-difference
operators on the torus H ⊂ G. The spherical DAHA for G Ln appears naturally [
43
] as
a quantization of the phase space of the trigonometric Ruijsenaars–Schneider system
[
27–29,44
] (also known as the relativistic version of the trigonometric Calogero–
Moser system), a many-body particle system with multiplicative dependence on both
positions and momenta. The phase space of this integrable system in turn has a
wellknown interpretation [
30,36
] in terms of the character variety of the torus, marked by
a distinguished “mirabolic” conjugacy class at one point. Namely it is identified with
a space of “almost-commuting” matrices, invertible matrices whose commutator lies
in a minimal conjugacy class (differing from a scalar matrix by a matrix of rank one).
In this section, we will explain how our theory of quantum character varieties
naturally produces the spherical DAHA when fed the torus T 2 marked by the quantum
mirabolic conjugacy class.
It is known that the spherical DAHA SHq,t may be obtained from the algebra Dq (G)
of quantum differential operators on G by quantum Hamiltonian reduction along the
quantum moment map μq : Oq (G) → Dq (G) at a certain equivariant two-sided ideal
It ⊂ Oq (G), depending on a parameter t . The ideal It is a canonical q-deformation
of the variety of matrices which differ from the scalar matrix t · id by a matrix of rank
at most one. These results give rise to multiplicative analogues of the relation between
the trigonometric Cherednik algebra (quantizing the trigonometric Calogero–Moser
phase space) and mirabolic D-modules, see [
22,25,34
].
Theorem 1.7 We have an isomorphism of algebras
( A(T 2)◦ /It )Uq glN ∼= SHq,t (G L N ),
in the following settings:
(1) When q is a root of unity [
46
],
(2) When q = e ( formal) for the Drinfeld-Jimbo category [
38
],
(3) For arbitrary q ∈ C× when N = 2 [
6
].
More precisely, we have slightly reformulated each result here, for a more uniform
presentation. Let us spell out the dictionary here, for the reader’s convenience. In [
46
],
an algebra of quantum differential operators on G L N × PN −1 is constructed, along
with a quantum moment map. In the notation of [
38
], this same algebra is denoted
Dq (Matd(Q)), for (Q, d) = •1 → N• , and is an important special case of the quiver
construction. We have isomorphisms of algebras,
N
Oq (G) ∼= Oq (•
) ∼= A Ann,
N
Dq (G L N ) =∼ Dq (•
) ∼= A(T 2)◦ ,
where each the first isomorphisms is clear from inspection of the defining relation,
and while the second are special cases of the main result of [
10
]. The PN −1 factor in
[
46
], and the extra vertex on the quiver in [
38
], each give rise to the deformed ideal
It , and so in each of the three cases of the theorem, one obtains the same quantum
Hamiltonian reduction algebra.
By Theorem 1.1, we may define a braided module category Mt = At -modA,
where At := Oq (G)/It comes equipped with a canonical quantum moment map,
as a quotient of Oq (G). Finally, the quantum Hamiltonian reduction computed in
those papers is precisely that which appears in Theorem 1.7. We therefore obtain the
following important corollary:
Corollary 1.8 Let A = Repq G Ln, M = Mt , S = (T 2, x ) the closed torus, with a
single marked point x colored by Mt . Then we have an isomorphism,
End(ix∗( At )) ∼= SHq,t (G Ln),
of the endomorphism algebra of ix∗( At ) as an object of (T 2,x)(A, Mt ), and spherical
DAHA SHq,t (G Ln).
Hence, we obtain a marked mapping class group-equivariant “global sections”
functor
:
(T 2,x)
M
→
(A, Mt ) → SHq,t (G Ln) -mod,
Hom(ix∗( At ), M )
from factorization homology of the marked torus to spherical DAHA-modules.
More generally, the category (T 2,x)(A, Mt ) provides a quantum version of the
category of mirabolic D-modules studied in [
4,25,42
] and others, of which
representations of spherical DAHA provide the “principal series” part.
This result gives a topological explanation of the existence of a quantum Fourier
transform on SHq,t (G Ln) leading to an action of the marked torus mapping class
group S L2(Z) by algebra automorphism. It also justifies the moniker “operator-valued
Verlinde algebra”, by which Cherednik first referred to his DAHA [
14,15
]: while the
Verlinde algebra is attached to T 2 by the 3d Witten-Reshetikhin-Turaev theory, we see
that the spherical double affine Hecke algebra is the affinization of the category attached
to a marked T 2 by the 4D theory, so that it obtains all the topological symmetries of
the torus from functoriality of the construction.
2 Factorization homology of surfaces
In this section we briefly review factorization homology of stratified n-manifolds,
following [
1,2,40
]. We will put special emphasis on the case n = 2, of surfaces—
possibly marked and/or with boundary—and with values in certain 2-categories Rex or
Pr, of k-linear categories (see [
10
] for a review of Rex / Pr as settings for factorization
homology): in this case, many of the structures demanded by the general framework
of factorization homology recover well-known structures in quantum algebra.
As in loc. cit. our main example will be the balanced tensor category Repq G:
this notation means we choose a reductive algebraic group G, a Killing form κ on
g = Li e(G), and consider either the category of finite-dimensional Uq (g)-modules,
when G is simply connected, or the corresponding braided tensor subcategory when
G is not semisimple. We do not recall a presentation of Uq (g) here, but rather refer to
e.g. [16, Sect. 9.1] for basic definitions.
2.1 Factorization homology of surfaces
Let s stand in for either framing (fr) or orientation (or). We denote by Mflds2 the
(∞, 1)-category, whose objects are (framed or oriented) 2-dimensional manifolds with
corners and whose morphisms spaces are the ∞-groupoids of (framed or oriented)
embeddings. We denote by Disks2 the full subcategory whose objects are arbitrary
(possibly empty) finite disjoint unions of R2. Each category is naturally symmetric
monoidal with respect to disjoint union.
Definition 2.1 A Disks2-algebra in an (∞, 1) symmetric monoidal category C, for
s ∈ { f r , or }, is a symmetric monoidal functor from Disks2 to C.
Remark 2.2 A Disk2f r -algebra (or rather the image of R2) is usually called an
E2algebra, or algebra over the little disk operad. Similarly a Disk2or -algebra is an algebra
over the framed little disk operad.
The data of A is completely determined, in the framed case, by the image A(R2)
of the generator R2, and a collection of morphisms A k → A (including k = 0,
which gives the unit map), and a well-known host of coherences. We abuse notation,
and denote both the functor and its value on the generator R2 by the symbol A. In the
oriented case, we have also to specify the “balancing” automorphism of the identity
functor. This corresponds to the loop, in the space of oriented diffeomorphisms of a
disc, which rotates θ degrees about the origin, for θ ∈ [0, 2π ].
Remark 2.3 Since our target is a (2,1)-category, functors from Mflds2 factors through
its “homotopy (2,1)-category”, i.e. the category whose Hom spaces are fundamental
groupoids of spaces of embeddings.
Our main case of interest is when C = Rex or Pr is a certain symmetric monoidal
2-category of k-linear categories with the Kelly-Deligne tensor product (see Sect. 4
below, and Sect. 3 of [
10
] for details). In this case, the data of an E2-algebra consists
of a braided tensor category structure on A(R2), which we denote simply as A, by
abuse of notation. The data of a Disk2or -algebra is identified with a balanced braided
tensor structure.
The factorization homology S A of an E2-algebra (resp, Disk2or -algebra) A on a
framed (resp, oriented) surface S is defined as a colimit,
S A = (Rc2o)likm→SA
k ,
over all framed (resp. oriented) embeddings of disjoint unions of discs into S, where the
(1-, and 2-) morphisms in the diagram are comprised of the tensor functors A k → A,
and their coherences (including the associativity and braiding isomorphisms).
In other words (in the case C = Rex / Pr), it is the universal recipient of functors
from A k labeled by collections of disjoint disks in S, and factoring through the
Disk2or -algebra structure on A, whenever a disk embedding factors through inclusions
of disks in a larger disk. Formally this colimit is expressed as a left Kan extension,
Disks2
A
Mflds2
−
A
C .
An important feature of factorization homology is that the empty set is regarded
as a surface, and has an initial embedding to any surface. This induces a map 1C
∅ A → S A, for any S. In the case C = Rex / Pr, we have 1C = Vect, so the
initial functor is determined by the the image of k ∈ Vect. This equips all braided
tensor categories appearing in the theory with their unit object, and equips factorization
homology of any surface S with a distinguished object, which we called in [
10
] the
“quantum structure sheaf,”
OS,A ∈
S
It follows that we can also calculate OA,S as the image of the unit in A under the map
A → S A associated to any disc embedding.
Factorization homology satisfies an important excision property. Given a
1manifold P, the factorization homology P×I A on the cylinder (with any framing)
carries a canonical E1 (associative) algebra structure from the inclusion of disjoint
unions of intervals inside a larger interval (i.e., we stack cylinders inside a larger
cylinder, see Fig. 1). Moreover the invariant of a manifold with a collared boundary
M is naturally a module over P×I A.
This structure allows us to describe the factorization homology of a glued (framed
or oriented) surface S = S1 P×I S2 as a relative tensor product:
S
A =
S1
A
P×AI
In the case of punctured surfaces studied in [
10
], we exploited excision for P = I ,
an interval, to compute factorization homology categories as categories of modules for
certain explicitly presentable “moduli algebras”. In the present paper, we will extend
these descriptions to closed surfaces, and this involves applying excision in the case
P = S1. For that case, we need to develop the theory of surfaces with marked points,
and an explicit description for the tensor structure on the annulus.
2.2 Marked points
Following [
2
], factorization homology for surfaces with marked points may be defined
similarly as for unmarked surfaces, via a Kan extension from a category of marked
discs to a category of marked surfaces.
We denote by Disks2,mkd the (∞, 1)-category whose objects are disjoint unions
of unmarked (framed or oriented) disks R2 and once-marked disks R20, and whose
morphism spaces are spaces of (framed or oriented) embeddings, which are moreover
required to send marked points bijectively to marked points.
The data of a symmetric monoidal functor Disk2f r,mkd → C is equivalent to an
E2algebra A (the restriction to unmarked disc) and an object M ∈ C assigned to a marked
disc, equipped with the structure of E2-module over A (the intrinsic operadic notion
of module for an E2-algebra). That is, M is equipped with a compatible collection
of functors M A k → M determined by choosing embeddings from the disjoint
union of one marked and k unmarked, disks back into the marked disk, satisfying a
collection of coherences. By analogy, we will call the image of the once marked disk
through a functor Disk2or ,mkd → C a Disk2or -module over the Disk2or -algebra image
of the unmarked disk.
Remark 2.4 The requirement that marked points map bijectively means that the empty
set is no longer initial in the category of marked surfaces. Allowing maps which are
only injective on marked points is equivalent to giving a pointing 1C → M with
no additional coherences, and agrees with the notion of locally constant factorization
algebra on the stratified space R2.
0
Similarly to ordinary factorization homology, a symmetric monoidal functor from
Disks2,mkd to some target category C is determined by its values A, and M on the
unmarked, and once-marked discs, respectively, together with a host of functors and
coherences between various tensor products of A and M. Let us therefore denote such
a functor by the pair (A, M).
Definition 2.5 The factorization homology of the pair (A, M) is the left Kan
extension,
Disk2or ,mkd
(A,M)
Mfld2or ,mkd
C .
−
(A, M)
As for ordinary factorization homology of surfaces, the definition by left Kan
extension implies a formula for the factorization homology of any marked surface as a
colimit,
(A, M) =
(S,X )
colim
(R2) k (R20) l →(S,X )
A
k
M
l ,
over all embeddings of unmarked or once-marked discs into S.
Just as for ordinary factorization homology, we have an excision property for
computing factorization homology of marked surfaces. Let
(S, X ) = (S1, X1) ∪P×I (S2, X2)
be the relative union of marked surfaces (S1, X1) and (S2, X2), along some (unmarked)
cylinder P × I . Then we have
(A, M) =
(S,X)
(A, M)
M1
A
P×I
(A, M).
M2
2.3 Boundary conditions
An important source of examples of markings of surfaces come from boundary
conditions in the topological field theory defined by A, or concretely from factorization
algebras on manifolds with boundary, extending A from the interior.
We denote by Disk2f r/otd,bdr y the (∞, 1)-category whose objects are disjoint unions
of unmarked (framed or oriented) disks R2 and half-spaces H = R × R≥0, and
whose morphism spaces are spaces of (framed or oriented) embeddings, which are
moreover required to respect boundaries. The data of a symmetric monoidal functor
Disk2f r,bdr y → C, i.e., of a factorization algebra on the stratified space H, is equivalent
to an E2-algebra A (the restriction to unmarked disc) and an object B ∈ C assigned to a
half-space, equipped with the structure of A-algebra: an algebra object in C equipped
with an E2-morphism z : A → Z(B) from A to the center
Z(B) = E ndB Bop (B)
of B (i.e. the pair (A, B) is an algebra over Voronov’s Swiss–cheese operad [
45
]).
Here the algebra structure on B comes from the inclusion of unions of half spaces into
half-spaces—i.e., B itself defines a one-dimensional factorization algebra valued in
C. The central action comes from the inclusion of a disc into the half space.
In the case C = Rex, this means we have a braided tensor category A, a tensor
category B, and a functor of braided tensor categories from A to the Drinfeld center
of B.
Two rich sources of A-algebras, hence boundary conditions, for A a braided tensor
category are:
(1) Categories of modules B = B -mod for commutative (i.e., braided or E2) algebra
objects B ∈ A.
(2) The category B = Repq B, of torus-integrable representations of the quantum
Borel subalgebra Uq (b+) ⊂ Uq (g) form a Repq G-algebra. This follows simply
from the fact that
R ∈ Uq (b+) ⊗ Uq (b−) ⊂ Uq (g) ⊗ Uq (g),
where R is the quantum R-matrix which controls the braiding on Repq G. Hence
R itself endows the forgetful functor Repq G → Repq B with the structure of
a central functor. More generally quantum parabolic subgroups define boundary
conditions.
Given a surface with boundary (S, ∂ S) we can perform factorization homology
(S,∂ S)
(A, B)
for the pair (A, B) following the same formalism as in the unmarked and marked
cases.
An A-algebra B is itself a one-dimensional factorization algebra and so can be
integrated on closed one-manifolds P → P B. In particular we have the trace /
cocenter / Hochschild homology category
tr (B) =
S1 B
(S1×R≥0,S1)
(A, B).
We thus see that tr (B) is naturally an S1×R A-module. When A is balanced, this is
identified with the annulus category U (A), and thus we find
Proposition 2.6 Let A denote a ribbon category and B an A-algebra.
(1) The trace tr (B) carries a natural structure of Disk2or A-module.
(2) Let (S, ∂ S) denote a compact surface with boundary ∂ S identified with n(S1),
and (S, {xi }) the closed surface with marked points obtaining by sewing in discs
along ∂ S. Then the factorization homology of A on S with boundary marked by
B agrees with that of A on S with points marked by tr (B):
(S,∂ S)
(A, B)
S,{xi }
(A, tr (B)).
The proposition (in the case of representations of a quantum Borel or parabolic
subgroup) allows us in particular to define parabolic versions of quantum character
varieties, quantizing moduli of parabolic local systems.
3 Braided module categories and surfaces with marked points
In this section, we identify the possible markings of points in factorization homology
of marked surfaces in terms of explicit algebraic data called braided module categories.
To begin, we recall that the factorization homology of the annulus inherits a natural
tensor structure, coming from “stacking annuli”:
Definition 3.1 The stacking tensor product on Ann A, denoted M , N
M
N , is:
A,
Ann
where iin and iout are as depicted below.
Remark 3.2 This same category carries a second tensor product, the pants, or
convolution tensor product. Let Pant s denote a twice punctured disc, let i1 and i2 denote
the inclusion of an annular boundary around the two punctures, and let iout denote the
annular boundary around the outside of the disc. Then we define:
TPants := io∗ut ◦ (i1
i2)∗ :
A
Ann
and the stacking tensor product is the pointwise tensor product of quasi-coherent
sheaves on the stack GG , while the pants tensor product is the convolution along the
diagram,
G G quot. G × G mult. G
G × G ←−−− G −−−→ G
.
The convolution tensor structure does not play a role in this paper, as it is the stacking
tensor product which features in the excision axiom.
There is a natural operadic notion of module for an En-algebra A, the notion of
En-module (which for n = 1 recovers the notion of bimodule rather than module over
an associative algebra), see [
31,41
] and [35] for a review. Intuitively an En-module
is an object placed at the origin in Rn and admitting operations from insertions of A
at disjoint points (or disks). This notion appears naturally in factorization homology,
as the data required to extend the theory to manifolds with marked points. There’s a
monadic description of En-modules over A as left modules over an E1-algebra U (A),
the enveloping algebra of A, which is in turn readily identified with the factorization
homology Ann A of A on the complement of the origin. We take this latter notion as
our definition and refer to [
31,35
] for comparisons with the operadic notion. We will
then give it a more algebraic description by reconciling it with the notion of a “braided
module category” [
8,23
], in the case C = Rex.
Definition-Proposition 3.3 ([
35
]) An E2-module for an E2-algebra A is a right module
over the annulus category Ann A, with tensor structure TSt .
In our setting, E2-algebras are identified with the notion of braided tensor
categories; in the same spirit, we first show that the notion of E2-modules over braided
tensor categories coincides with the notion of braided module categories defined as
follows:
Definition 3.4 ([
8
]) Let A be a braided tensor category, with braiding σ . A (strict2)
braided module category over A is a (strict) right A-module category M equipped
with a natural automorphism, E , of the action bifunctor
satisfying the following axioms for all M ∈ M, X , Y ∈ A:
⊗ : M × A −→ M
E M⊗X ,Y = σX−,1Y E M,Y σY−,1X
and
E M,X ⊗Y = E M,X E M⊗X ,Y σY ,X σX ,Y .
E M,X ⊗Y = E M,X σX−,1Y E M,Y σX ,Y .
(3.1)
(3.2)
(3.3)
Remark 3.5 In the presence of (3.1), equation (3.2) is equivalent to the following
version of the Donin–Kulish–Mudrov equation [
17
]
This in turn implies the reflection equation, and hence that braided module
categories give rise to representations of the braid group of the annulus. In fact
equation (3.1) alone implies the reflection equation, but both are needed to describe the
full E2-module structure.
Remark 3.6 Axioms (3.1), (3.2) differ slightly from those in [
8
]: the latter characterizes
modules over the factorization homology of S1 ×R equipped with the product/cylinder
framing, while E2-modules are rather characterized as modules over the factorization
homology of S1 × R equipped with the blackboard (annulus) framing. As there are an
integer’s worth of framings on S1 × R, hence there are an integer’s worth of alternative
notions of braided module category.
When A is balanced, however the corresponding axioms are equivalent: indeed,
if θ is the balancing and E satisfies the axioms stated above, then a straightforward
computation shows that E −1(id ⊗θ −1) satisfies the axioms of [
8
]. We note that a
similar phenomenon has appeared in the definition of the elliptic double of [
5
], where
there are many possible definitions, which coincide in the case of a ribbon Hopf algebra.
Here, in order to simplify the presentation, we stick to the balanced assumption and
do not differentiate between these different framings.
In order to identify the different notions of module category we will use a different
description of the annulus than that which features in [
10
]. Rather than cutting it into
two half-annuli, as we did there, we will make a single vertical cut at the top of the
annulus and see the annulus as being obtained by self-gluing along this cut (see Fig. 2).
Hence let Y be the manifold represented in Fig. 2. We stress the fact that for the
gluing to make sense we need to regard it as a stratified manifold, and as such it
2 This assumption is made only to simplify the exposition: the non-strict axioms simply involve inserting
associators in the obvious places in equations (3.1) and (3.2).
Fig. 3 The A-bimodule Y A is obtained from the regular bimodule by precomposing the right action by
the tensor functor (i d, σ 2) : A → A
is not equivalent to the standard framed disc. Yet, its interior Y˚ , i.e. the manifold
obtained from Y by forgetting the stratification, is equivalent as a framed manifold to
the standard disc, hence we have an equivalence of underlying categories,
Y
A
˚ A
Y
A
but the A-actions are different from the standard one. The A-bimodule action on Y A
is induced by the embedding i R i0 i L : R2 Y˚ R2 → Y depicted in the left hand
side of Fig. 2, i.e. it is the functor,
(i R
i0
i L )∗ : A
Y
A
A →
Y
A,
The double braiding σ 2 induces a non-trivial tensor structure on the identity functor
of A. We denote by AA(id,σ 2) the twist of the regular bimodule category by this
autoequivalence, i.e. the A-bimodule category whose underlying left module is the regular
left A-module, but where the right action (in particular, its associativity constraint) is
precomposed with (i d, σ 2). In Fig. 3, we demonstrate an equivalence between Y A
and this twisted bimodule.
To recover the factorization homology of the annulus from the bimodule Y A, we
need to recall the notion of balanced functors of bimodules, and the resulting notion
of the trace of a bimodule. To this end, let M be an A − A-bimodule category. Then
a functor F : M → E is called balanced if there is a natural isomorphism
F (m ⊗ X ) → F ( X ⊗ m)
satisfying a natural coherence condition (see e.g. [
32
]).
Definition 3.7 The trace tr M, of an an A-A-bimodule category M, is the
Rexkcategory defined uniquely by the natural equivalence,
Rexk[tr M, E ]
BalA(M, E ).
Remark 3.8 If M, N are right and left A-module categories, then M
bimodule and clearly
N is an
AOn the other hand, an A-bimodule is the same as an A
show that
Arev-module and one can
tr (M
N )
M
A N .
tr M
A
A Arev M
where A is given its natural A-bimodule structure (i.e., we recover the standard notion
of Hochschild homology of a bimodule).
Ann A is the trace in the sense of Definition 3.7 of the
Proof It is clear that constructing the annulus by self-gluing from the manifold showed
on Fig. 2 is the same as gluing both ends to a disk with two marked intervals and its
standard framing, which corresponds to the regular bimodule A. Hence by the excision
property we have:
Ann
A
A A(id,σ 2)
A Arev A
tr
AA(id,σ 2) .
We can now prove:
Proposition 3.10 Let A be a braided tensor category, B a tensor category. Then tensor
functors Ann A → B are naturally identified with pairs (F , ν) where F is a tensor
functor from A to B, and ν is a natural automorphism of F satisfying3:
and
3 Here and in the proof we abuse notation and write σX,Y instead of F(σX,Y ).
idX ⊗νY = σX−,1Y (νY ⊗ idX )σY−,1X
νX⊗Y = (νX ⊗ νY )σY ,X σX,Y
(3.4)
(3.5)
Proof Denote by and the left and right A-actions on itself described above. The
description of Ann A as a coequalizer of those actions implies that functors out of it
to a target category B are naturally identified with functors F : A → B equipped with
a natural isomorphism
ηX,Y : F (X ⊗ Y ) = F (X
Y ) → F (Y ⊗ X ) = F (Y
X )
satisfying the following coherence condition
σZ,Y σY ,Z ηX,Y ⊗Z = ηZ⊗X,Y ηX⊗Y ,Z
In order to characterize the monoidal structure on Ann A, we need to upgrade this
coequalizer as the coequalizer of a diagram in tensor categories. Recall that any tensor
functor F : A → B turns B into a left A-module using the following composition
A
F id
B −−−→ B
m
B −→ B
where m is the multiplication of B and the associativity constraint of the module
structure is given by the monoidal structure on F . This also turn B into a right module
by using the opposite multiplication instead. We note that both A actions at hand are
of this form: the right action is just induced by the identity functor and the left action
by the identity functor with monoidal structure given by the double braiding. Since
the multiplication of A also carries a natural monoidal structure, this turns the maps
involved in the defining diagram of Ann A into tensor functors.
Now, (strict) monoidal functors out of Ann A to a monoidal category B can be
characterized as strict monoidal functors F : A → B equipped with a cyclic structure
η as above for which η is monoidal. This leads to the following identity:
ηX⊗W ,Y ⊗Z σY ,W = σZ,Y σY ,Z σX,Z ηX,Y ηW ,Z
The coherence condition implies that ηX,1 = idX . Hence define a natural
automorphism of F by νX := η1,X . Setting X = Z = 1 gives
so that η can be recovered from ν. For W = Y = 1 this gives
Together those equations implies (3.4). Finally, setting X = W = 1 leads to
ηW ,Y σY ,W = νY
ηX,Z = σX,Z νZ .
νX⊗Z = σZ,Y σY ,Z νX νZ .
which up to relabelling is exactly (3.5).
Hence, we have:
Theorem 3.11 Let A be a braided tensor category. Then E2-modules over A are naturally identified with A-braided module categories in the sense of [8].
Proof A right module M over Ann A is characterized by the tensor functor Ann A →
End(M) given by X → − ⊗ X . It means that M has to be an A-module and that
one has to provide the functor A → End(M) with a natural automorphism satisfying
the axioms above. This is straightforward to check that this turns M into a braided
module category.
3.1 The oriented case
Recall that factorization homology of framed surfaces descends to an invariant of
oriented surfaces provided A is balanced, i.e. that A is endowed with an automorphism
θ : idA → idA of the identity functor of A, satisfying the coherence relation,
θV ⊗W = σW ,V ◦ σV ,W ◦ (θV ⊗ θW ).
In the same way, one can ask what additional structure is needed on a braided
module category M over a balanced braided monoidal category A in order to obtain
invariants of oriented marked surfaces. We have:
Theorem 3.12 Let A denote a braided tensor category and let M ∈ Rex.
(1) Given a braided module category structure on M, the additional structure of a
Disk2or -module extending A and M consists, first of all of a balancing on A,
and secondly of a “balancing automorphism” φM : idM → idM of the identity
functor on M, satisfying the coherence:
φM⊗X = E M,X ◦ (φM ⊗ θX )
(3.6)
We refer to a braided module category M, equipped with a φ , as a “balanced
M
braided module category”.
(2) Suppose that A is a balanced braided tensor category, and that M is a braided
module category for A. Then M admits a canonical structure of a balanced
braided module category.
Proof The first part is clear from the picture: φ is the automorphism of the identity
functor of M induced by the rotation of a marked disc inside a larger one.
If A is balanced, then Ann A is independent of the framing and in particular comes
equipped with an automorphism of the identity functor ψ coming from the loop, in
the space of oriented diffeomorphisms of the annulus, which rotates θ degrees about
the origin, for θ ∈ [0, 2π ]. Let F = Fbd : A −→ Ann A be band tensor functor
defined in Fig. 4, and ν the automorphism of F as in Proposition 3.10. It is again a
direct check that for all X ∈ A,
Fig. 4 Left: the tensor functor Fbd : A → Ann A is induced by an inclusion of a disc into a small radial
band in the annulus. Right: the tensor structure induced by a commutative diagram, up to isotopy
where we identify F (X ) with F (X ) ⊗ 1 Ann A.
By Proposition 3.10, a braided module structure E on M is the same as a
factorization of the functor
A −→ End(M)
A −→
Ann
A −→ End(M )
and E is defined as the image of ν through the second functor. Taking the image
of (3.7) through the second functor gives the desired balanced structure on M.
Remark 3.13 We note that the balancing asserted in (2), while canonical, is not unique.
There may be many different balancings on a given braided module category.
4 Reconstruction theorems
In [
10
], we developed a framework to describe the gluing of surfaces along intervals
in their boundary, using monadic techniques: the factorization homology for a surface
with a marked interval in the boundary obtained the structure of an A-module category,
and we realized these A-module categories as the categories of modules for explicit
algebras AS in A. In this section, we will develop some of the general algebraic tools
we will need to glue along circles, rather than intervals, in the boundary of a surface.
Namely, the pointwise tensor structure from Fig. 4 defines a dominant tensor functor
from A to Ann A, and we wish to apply monadic techniques to understand the resulting
structures algebraically.
Assumptions We will typically work with the (2, 1)-categories Rex of essentially
small finitely cocomplete k-linear categories, right exact functors and their natural
isomorphisms, and the (2, 1) category Pr of compactly generated presentable
categories with compact and cocontinuous functors and their natural isomorphisms. These
each carry the so-called Kelly-Deligne tensor product, and are in fact equivalent to
one another as symmetric monoidal (2, 1)-categories: the functors ind and comp, of
ind-completing a Rex category to a Pr category, and taking the Rex subcategory of
compact objects of an Pr category, are mutually inverse equivalences. For orientation,
let us remark that small abelian categories are in particular Rex, while Grothendieck
abelian categories are in particular Pr.
By a tensor (or braided tensor) category in Rex / Pr, we will mean simply an
E1(or E2-) algebra A in Rex / Pr. We will typically assume that A is rigid, i.e. that all
(compact) objects are left and right dualizable. This categorical framework is discussed
in detail in [
10
], to which we refer the reader for complete definitions.
4.1 Reconstruction from tensor functors
Let A and B be tensor categories in Pr and F : A → B be a tensor functor. Suppose
that A is rigid, and assume the 1B is a pro-generator for the A-module structure
on B induced by F 4. The definition of a pro-generator implies that F = act1 has
B
a cocontinuous right adjoint which is also faithful, i.e. F is dominant. Hence, by
applying Theorem 4.5 from [
10
] to B as a A-module category, we see that B admits
a simultaneous description, both as right-, and as left- End(1B)-modules in A, where
we recall that End(1B) ∼= F R (1B). In this section, we extend this description to
encompass the tensor structure on B as well.
Theorem 4.1 We have:
(1) Under this description F identifies with the free module functor V → End(1B) ⊗
V , and F R identifies with the forgetful functor back to A, where End means
internal endomorphism of B as an A-module.
(2) The right and left End(1B)-module structures on F R (b), for any b ∈ B, are
related by the isomorphisms:
End(1B) ⊗ F R (b) ∼= F R (1B ⊗ b) ∼= F R (b ⊗ 1B) =∼ F R (b) ⊗ End(1B).(4.1)
These actions commute, and we obtain a faithful tensor functor,
F R : B → End(1B)-bimodules in A.
(3) Moreover F R becomes a tensor functor, when we equip the category of
End(1B)bimodules in C with the relative tensor product of bimodules:
M
4 Note that it is a pro-generator for the left A-action induced by F if and only if it is for the right one.
Proof For Claim (1), we need a natural isomorphism F R F (V ) ∼= End(1B) ⊗ V .
Applying the tensor structure and the adjunction counit, we obtain natural
isomorphisms:
Hom(C , F R F (V )) ∼= Hom(F (C ), F (V ))
∼= Hom(∗ F (V ) ⊗ F (C ), 1B)
∼= Hom(F (∗V ) ⊗ F (C ), 1B)
∼= Hom(F (∗V ⊗ C ), 1B)
∼= Hom(∗V ⊗ C , F R (1B))
∼= Hom(C , F R (1B) ⊗ V ).
Hence by Yoneda’s lemma, we have a natural isomorphism F R F (V ) ∼= F R (1B) ⊗ V ,
as desired.
Claim (2) is clear: the End(1B)-actions on F R (b) are given in terms of the adjunction
data (F , F R ); the isomorphisms (4.1) are natural in b, and interchange left and right
modules in the adjunction.
For Claim (3), we note that the category of F R (1B)-bimodules is generated under
colimits by the free bimodules, so that if we have such an isomorphism for the free
bimodules, it necessarily induces the same natural isomorphism for all bimodules.
Hence, we may restrict to the case that M and N are of the form F (m), F (n), for some
m, n ∈ B. In that case, applying F R F to the obvious idenity,
m ⊗ n = colim
actR
m ⊗ 1A ⊗ n ⇒ m ⊗ n ,
actL
gives
F R (F (m) ⊗ F (n)) = F R
colim
actR
F (m) ⊗ 1B ⊗ F (n) ⇒ F (m) ⊗ F (n)
actL
.
While we cannot commute F R past the colimit, we have the canonical comparison
map:
colim
actR
F R (F (m) ⊗ 1B ⊗ F (n)) ⇒ F R (F (m) ⊗ F (n))
actL
→ F R
colim
actR
F (m) ⊗ 1B ⊗ F (n) ⇒ F (m) ⊗ F (n)
actL
.
Moreover, we have F R (F (m) ⊗ 1B ⊗ F (n)) ∼= F (m) ⊗ F R (1B) ⊗ F (n), because the
right adjoint F R to the A-bimodule functor F is canonically a A-bimodule functor
whenever A is rigid. Finally, we see that the comparison map is in fact an isomorphism,
because F is the free bimodule functor.
Having described the tensor structure on B monadically through A, we now turn to
describing algebra objects, and hence module categories, for B, monadically in terms
of A. We have:
Proposition 4.2 The equivalence B
End(1B) -modA extends to an equivalence:
{Algebras in B}
Algebras in A, equipped with an
algebra homomorphism from End(1B) .
Proof Given an algebra object b in B, its image F R (b) in A receives a canonical algebra
homomorphism F R (1B) → F R (b), induced by the unit homomorphism 1B → b,
through the lax tensor structure on F R . This provides a functor in the forward direction.
Conversely, given an algebra in B equipped with a homomorphism from F R (1B),
we make it an algebra in the category of F R (1B)-modules via this homomorphism;
this clearly endows it with the structure of an algebra object in F R (1B)-modules. This
provides a functor in the reverse direction.
The two functors we have constructed are mutually inverse for tautological reasons:
the unit homomorphism in the forward direction, and the F R (1B)-module structure
in the reverse direction are simply equivalent data.
Moreover, we can describe the B action on one of its module categories monadically
through A:
Theorem 4.3 Let M be a B-module category, and let M ∈ M be a progenerator for the induced A-action. Recalling the equivalences of A-modules,
B
modA −End(1B),
M
End(M ) -modA,
and the canonical algebra homomorphism,
ρ : End(1B) → End(M ),
we have:
(1) The action of any b ∈ B on any N ∈ M is given by:
N
b → b
(2) The A-progenerator M is also a progenerator for the B-action.
(3) The functor M = EndB(M ) -modB → EndA(M ) -modA induced by ρ is an
equivalence of B-module categories.
Proof The first claim is a direct application of Proposition 4.1. For the second claim,
we recall that, to say M is a pro-generator, is to say that the functor actMR : M → B is
conservative, and co-continuous. It follows from our assumptions that the composite
functor F R ◦ act MR is conservative and co-continuous, as the action of A on M is
obtained by pull-back, and M is assumed to be a A pro-generator. By assumption
F R : B → A is conservative and co-continuous. It is an easy exercise that conservative
and co-continuous functors themselves reflect conservativity and co-continuity. The
third claim follows from monadicity for base change, Corollary 4.11 in [
10
].
And finally, we can compute relative tensor product of module categories, simply
as categories of bimodules:
Corollary 4.4 Fix A, B and F as in Proposition 4.2. Let M and N be left and right
B-module categories, with A-progenerators m and n, respectively. Then we have an
equivalence of categories,
M
B
N
(End(m) − End(n)) -bimodEnd(1B) -modA .
Proof This is just an application of Theorem 4.9 from [
10
], and Proposition 4.3 above.
4.2 Reconstruction for the annulus category
In this section, we apply the reconstruction techniques of the preceding section to the
setting of factorization homology for braided module categories, to give a proof of
Theorem 1.1. To begin with, we consider the inclusion j0 : D2 → Ann, given by
including a small disk D2 into an annulus along some small band, as depicted in Fig. 4.
Definition 4.5 The “band” tensor functor Fbd : A → Ann A is the functor ( j0)∗,
induced by functoriality of factorization homology, with tensor structure induced by
the commuting-up-to-isotopy diagram of inclusions, depicted in Fig. 4.
With all of this framework in place, we turn to the proof of Theorem 1.1.
Proof of Theorem 1.1 We shall apply the results of Sect. 4 to the special case that A is
a braided tensor category in Rex, and
D =
Ann
A
FA -modA,
F = Fbd .
It follows from Theorem 4.16 of [
10
] that OA is a pro-generator for the A-action,
and that Fbd =∼ actOA . Hence, by Theorem 4.1, Fbd is naturally isomorphic to the
free module functor V → FA ⊗ V , and FbRd : Ann A → A identifies simply with the
forgetful functor from FA -modA → A.
For (1), the quantum moment map is that constructed in Proposition 4.2. For (2),
Theorem 4.5 from [
10
], combined with Proposition 4.2, give equivalences:
M
EndA(M ) -modA
End (M ) -mod
A
Ann A
,
where the latter is equipped with the algebra structure coming from the quantum
moment map. Finally, (3) follows the same proof as Part (3) of Proposition 4.1.
Unpacking the isomorphism of Theorems 4.1 and 4.3 in this case, we have the
following corollaries:
Corollary 4.6 Any left FA-module has a canonical right module structure. The left and
right action are related using the “field goal” transform τm : FA ⊗ m → m ⊗ FA:
τm =
m
FA
FA
m
.
For any M , N ∈ Ann A FA -modA, the pointwise tensor product M
by the relative tensor product,
N is given
M
N ∼= M ⊗ N := colim
FA
actR
M ⊗ FA ⊗ N ⇒ M ⊗ N ,
actL
where M is made into a right FA-module by the field goal transform.
Corollary 4.7 We have an equivalence of categories,
Algebras in
A
Ann
aAllggeebbrraashionmAom,eoqrpuhipispmedfrwoimthFaAn .
Moreover, given a module category M = A -modA, for an algebra A ∈ A equipped
with an algebra morphism ρ : FA → A, the action of Ann A on M is given by:
A -modA
FA -modA −→ A -modA
V M −→ V ⊗FA M
where FA-acts on V via ρ.
Corollary 4.8 Let M and N be a left and a right module category over Ann A with
progenerators M and N , respectively. Then there is an equivalence of categories:
M
N
A
Ann
(End(M ) − End(N )) -bimodFA -modA .
4.3 Braided module structure
It follows from the above that if B is an algebra in A, every algebra morphism
ρ : FA → B turns B -mod into a module category over Ann A. Hence ρ should
correspond to a braided module structure on the category B -modA in the sense of
Definition 3.4. In this section we make this structure explicit; the construction which
follows can be interpreted as a generalisation of [
17
].
Let B be an algebra in A and an automorphism of the action functor
We begin by constructing a morphism of underlying objects,
B -modA ×A → B -modA .
ρ : FA −→ B,
as follows. Using the definition of FA as a co-end, it suffices to define ρ compatibly
on each V ∗ ⊗ V , as below:
1B ⊗id⊗2 B,V ∗ ⊗id
ρ |V ∗⊗V : V ∗ ⊗ V −−−−−→ B ⊗ V ∗ ⊗ V −−−−−→ B ⊗ V ∗ ⊗ V −id−⊗−→ev B
It is easily checked that this system of maps descends to FA.
Theorem 4.9 The morphism ρ is an algebra homomorphism if and only if satisfies
equation (3.3):
M,V ⊗W = σV−,1W
M,W σV ,W M,V
Proof Let m B denote the multiplication of B. On the one hand, we consider the
composition, m B ◦ (ρB ⊗ ρB ):
(V ∗ ⊗ V ) ⊗ (W ∗ ⊗ W ) → (B ⊗ V ∗ ⊗ V ) ⊗ (B ⊗ W ∗ ⊗ W )
B,V ∗ ⊗id ⊗ B,W∗ ⊗id
−−−−−−−−−−−−−→ (B ⊗ V ∗ ⊗ V ) ⊗ (B ⊗ W ∗ ⊗ W ) → B ⊗ B −m−→B B.
(4.2)
(V ∗ ⊗ V ) ⊗ (W ∗ ⊗ W ) −σ−V −∗−⊗V−,−W→∗ (W ∗ ⊗ V ∗) ⊗ V ⊗ W
B,W∗⊗V ∗ ⊗id
→ B ⊗ (W ∗ ⊗ V ∗) ⊗ V ⊗ W −−−−−−−−→ B ⊗ (W ∗ ⊗ V ∗) ⊗ V ⊗ W → B.
(4.3)
In order to simplify the computation, we precompose each side by σV−∗1⊗V ,W ∗ . Then
the homomorphism equation (4.2) = (4.3) can be expressed as follows:
L H S
B
W ∗
V ∗
V
W
W ∗
V ∗
V
W
W ∗
V ∗
V
W
W ∗
V ∗
V
W
Since is B-linear, B,− commutes with the action of b on itself by left multiplication.
Thus, beginning with the LHS, we may slide the rightmost instance of over the
multiplication, at which point the rightmost unit disappears. We have:
Clearly, this equals the RHS if, and only if, we have
W ∗
V ∗
B
W ∗
V ∗
W ∗
V ∗
B
W ∗
V ∗
Conjugating the above by σW ∗,V ∗ , and replacing V ∗, W ∗ with V , W gives equation
(3.3). Therefore, compositions (4.2) and (4.3) coincide if and only if (3.3) holds.
theWreearweialllsaoppcalynoTnhiecoalremmap4s.9 in the following particular case: by definition of FA,
V ⊗ ∗V −→ FA
where ∗V is the right dual of V . For M ∈ FA -modA and V ∈ A, let LA ∈ Aut (M ⊗
V ) be the operator defined by
id⊗2 ⊗coevR σM,V ⊗∗V ⊗id
M ⊗ V −−−−−−−→ M ⊗ V ⊗ ∗V ⊗ V −−−−−−−→ V ⊗ ∗V ⊗ M ⊗ V
−→ FA ⊗ M ⊗ V −→ M ⊗ V
where the last map is the action of FA on M .
iPtrsoeplfoisnidtiuocne4d.1b0y LUAndiserththeeidiednetnittiyfi. cIantipoanrtuicsuedlairnthTihseiosraenma4lg.9e,btrhaemmoarpphfrisomm, FheAnctoe
LA satisfies equation (3.3).
Proof We have:
ιV
(ρOA )|V ∗⊗V =
= ιV
1A
V ∗
V
Corollary 4.11 Let B be an algebra in A. If a braided module structure on B -mod,
then ρ is an algebra morphism FA → B. Conversely, if ρ is an algebra morphism
FA → B, then using ρ FA acts on any B-modules M , hence LA acts on M ⊗ V for
V ∈ A and this turns B -mod into a braided module category.
5 Closed surfaces, markings and quantum Hamiltonian reduction
We now close up punctured surfaces to describe quantum character varieties for closed,
possibly marked, surfaces. For simplicity, we will begin with the unmarked situation.
Fixing an inclusion D2 → S, and the corresponding boundary inclusions of the
annulus into S◦ = S\D2 and D2 as boundary annuli, excision gives a canonical
equivalence,
S
A
S◦
A
A
Ann
A.
S◦
A
Fb∗d
S◦
Here A = D2 A is regarded as a braided module category over itself.
We now proceed to compute these invariants explicitly via quantum Hamiltonian
reduction. Recall that the algebra AS◦ is identified with act ORS◦ (OS◦ ) ∈ A, where A
acts on S◦ A via insertion at some interval in the boundary annulus of A. We have
A-module category equivalences,
where Fbd : A → Ann A is the tensor functor from Definition 4.5, induced by the
chain of inclusions depicted in Fig. 5.
This means we are precisely in the situation of Sect. 4.2. We have canonical quantum
moment maps,
μ : FA → AS◦ ,
: FA → 1A,
which are A-algebra homomorphisms realizing the braided module structures on
S◦
A
AS◦ -modA,
D2 A
A
1A -modA,
respectively, and moreover we have equivalences of Ann A-module categories,
S◦
A
AS◦ -modFA -modA ,
D2 A
1A -modFA -modA
between the factorization homology of S◦ (resp, D2), as module categories for the
annulus category by stacking, and the categories of AS◦ -modules in A (resp,
1Amodules in A), equipped in each case with compatible actions of FA.
Remark 5.1 The map μ is a generalization of the “quantum moment maps” studied in
[
38,46
]; it quantizes the monodromy map,
ChG (S\D2) → GG
.
As an application of Corollary 4.8, we obtain:
Theorem 5.2 The category attached to a closed surface is given as ( AS◦ ,
1A)bimodules in the annulus category,
S
A
( AS◦ -mod- 1A)FA -modA .
Fig. 5 The action of A on S◦ A is obtained by pulling back the AnnA-action, through the tensor functor
Fbd : A → Ann A
5.1 Marked points
Let X ⊂ S denote a finite set of points. Let us fix braided A-module categories
Mi attached to each points xi ∈ X , and let X = Dx1 ∪ · · · ∪ Dxr denote a tubular
neighborhood of X , consisting of the disjoint union of some discs Dxi containing each
point xi , and let S◦ = S\X . This data defines an invariant,
(S, X ) →
(A, {M1, . . . , Mr })
(S,X)
of surfaces with marked point x labeled by M. Applying excision gives an
equivalence:
(A, {M1, . . . , Mr })
(S,X)
A
S◦
I (M1
· · ·
Mr ) .
Now let us assume furthermore that each Mi is given as the category of modules
for an algebra Ai ∈ A, i.e. that
Giving such a presentation is equivalent to giving an A-progenerator Mi ∈ Mi as an
A-module category, by taking Ai = End(Mi ). It follows from Theorem 1.1 that each
Ai canonically receives a quantum moment map,
A
Ann
Mi = Ai -modA .
μi : FA → Ai ,
such that the Ann A-module action is identified with the relative tensor product over
μi .
Applying Corollary 4.8, we obtain:
Theorem 5.3 The factorization homology of the marked surface (S, X ), with braided
module categories Mi = mod- Ai attached to each xi is equivalent to the category,
(S,X)
(A, {M1, . . . , Mr })
( AS\X -mod-( A1
· · ·
Ar )F r -modA ,
A
of bimodules in F⊗Ar -modA for the pair of algebras AS\ Di2 and A1
· · ·
Ar .
5.2 The functor of global sections
Let us now turn to computing the endomorphism algebra of the distinguished object
OA,S ∈ S A. To this end, we recall first of all that we have an isomorphism,
OA,S =∼ OA,S◦
OA,D2 .
A
Ann
(5.1)
More generally, for any braided module category M, and any N ∈ M, we have,
ix∗(N ) ∼= OA,S◦
A
Ann
N .
Theorem 5.4 We have an isomorphism of algebras,
End(OA,S ) =∼ HomA
1A, AS\D2 F⊗A 1A ,
between the endomorphism algebra of the distinguished object and the Hamiltonian
reduction algebra.
More generally, if S is a surface with a point marked by a braided module category
M, then, for any N ∈ M, we have an isomorphism of algebras:
End(ix∗(N )) ∼= HomA
1A, AS\D2 F⊗A End(N ) ,
where the relative tensor product is taken over the canonical homomorphism ρ :
FA → N given by Proposition 4.2.
Proof The proof is based on a concrete description of Hom spaces between pure tensor
products of objects in relative tensor product categories, which was proved in [
21
].
We recall:
Proposition 5.5 ([21]) Given a right module category M, and a left module cate
gory N over a rigid tensor category C, and m, m ∈ M, n, n ∈ N , we have an
isomorphism,
Hom
M C N (m
C
n ) ∼= HomC (1C , Hom(m, m ) ⊗ Hom(n, n )).
Combining the above proposition with the tensor product decomposition of equation
(5.1), we have an isomorphism,
C
End S A(i∗(N )) ∼= End S A ⎝⎜ OA,S\D2
A
Ann
N ⎟
⎠
∼= Hom
∼= Hom
Ann A
Ann A
FA, End Ann A(OA,S\D2 )
End
Fbd (1A), End Ann A(OA,S\D2 )
∼= HomA 1A, FbRd End Ann A(OA,S\D2 )
End
Ann A
End
Ann A
(N )
Ann A
(N )
(N )
∼= HomA
1A, AS\D2 F⊗A EndA(N ) ,
Remark 5.6 In the case A = Rep G, the distinguished object can be identified with
the structure sheaf of the character stack. Hence the functor = Hom(OA,S , −) can
be viewed as a “global sections functor” on the quantized character stack S Repq G.
We note that the procedure prescribed in Theorem 5.4, of tensoring with the trivial
module along the quantum moment map, and then taking invariants, is precisely the
procedure of quantum Hamiltonian reduction.
Acknowledgements We would like to thank Pavel Etingof and Benjamin Enriquez for sharing their ideas
concerning elliptic structures on categories. We’d also like to thank John Francis, Greg Ginot, Owen
Gwilliam, and Claudia Scheimbauer for discussions about factorization homology.
The work of DBZ and DJ was partly supported by NSF grant DMS-1103525 and ERC Starting Grant
637618. DBZ and DJ would like to acknowledge that part of the work was carried out at MSRI as part of
the program on Geometric Representation Theory.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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