Abstract
We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {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 {mathcal {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 {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.
Highlights
Let S denote a topological surface and G a reductive group
We show in Theorem 3.11 that codimension two defects ( Ann A-modules) are identified with braided module categories over A, in the same way that the unmarked disc is assigned A
Factorization homology provides a general mechanism to construct invariants of nmanifolds starting from algebras A over the En operad—i.e., objects which carry operations labeled by inclusions of disks into a larger disk
Summary
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). [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. 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. (3) By their construction—as equivariant quotients of the reflection equation algebra—quantizations of conjugacy classes carry canonical quantum moment maps
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