Quantum character varieties and braided module categories

TL;DR

The paper develops a quantum generalization of character varieties for surfaces by encoding them as category-valued invariants $\\int_S \\mathcal{A}$ arising from factorization homology with a braided tensor category $\\mathcal{A}$. It identifies codimension-two data (marked points and boundaries) with braided module categories and introduces quantum moment maps $\\mu_M: \\mathfrak{F}_{\\mathcal{A}} \\to A_{M}$ that govern gluing via quantum Hamiltonian reduction. The torus and its quantum symmetries are linked to adjoint-equivariant quantum $\\mathcal{D}$-modules via $\\mathcal{D}_q(G/G)$-mod, and, in the $G = GL_n$ case, the endomorphism algebras of certain objects yield the spherical double affine Hecke algebra $\\mathbb{SH}_{q,t}$, providing a conceptual bridge between 4D TFT data and DAHA representations. The framework supports quantum character sheaves and a topological interpretation of operator-valued Verlinde algebras, with implications for mirabolic $\\mathcal{D}$-modules and loop-group categorifications. Overall, the work offers a cohesive, diagrammatic approach to quantizing character varieties and their representations through factorization homology and braided module theory.

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 {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em 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 $\mathcal A=\operatorname{Rep_q} G$ with the category $\mathcal D_q(G/G)-\operatorname{mod}$ 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 (DAHA) $\mathbb{SH}_{q,t}$.

Figures (5)

• Figure 1: The inclusion of two annuli into a third induces the stacking tensor structure on $\int_{Ann}\mathcal{A}$ by functoriality.
• Figure 2: Left: the stratified manifold $Y$. Right: the inclusions of discs inducing the $\mathcal{A}$-bimodule structure on $\int_Y\mathcal{A}$. The annulus is obtained by gluing along the boundary intervals.
• Figure 3: The $\mathcal{A}$-bimodule $\int_Y\mathcal{A}$ is obtained from the regular bimodule by precomposing the right action by the tensor functor $(id,\sigma^2):\mathcal{A}\rightarrow \mathcal{A}$.
• Figure 4: Left: the tensor functor $F_{bd}:\mathcal{A}\to\int_{Ann}\mathcal{A}$ 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.
• Figure 5: The action of $\mathcal{A}$ on $\int_{S^\circ}\mathcal{A}$ is obtained by pulling back the $\int_{Ann}\!\mathcal{A}$-action, through the tensor functor $F_{bd}:\mathcal{A}\to\int_{Ann}\mathcal{A}$.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4: Braided $G$-categories and loop group categories
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Definition 2.1
• Remark 2.2