The character field theory and homology of character varieties

We construct an extended oriented $(2 + \epsilon)$-dimensional topological field theory, the character field theory $\mathcal{X}_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of surfaces. It is a model for a dimensional reduction of Kapustin-Witten theory ($\mathcal{N} = 4 d = 4$ super-Yang–Mills in the GL twist), and a universal version of the unipotent character field theory introduced by two of the authors. Boundary conditions in $\mathcal{X}_G$ are given by quantum Hamiltonian $G$-spaces, as captured by de Rham (or strong) $G$-categories, i.e., module categories for the monoidal dg category $\mathcal{D}(G)$ of $\mathcal{D}$-modules on $G$. We show that the circle integral $\mathcal{X}_G (S^1)$ (the center and trace of $\mathcal{D}(G))$ is identified with the category $\mathcal{D}(G/G)$ of “class $\mathcal{D}$-modules”, while for an oriented surface $S$ (with arbitrary decorations at punctures) we show that $\mathcal{X}_G (S) \simeq H^{BM}_{\ast} (\operatorname{Loc}_G (S))$ is the Borel–Moore homology of the corresponding character stack. We also describe the “Hodge filtration” on the character theory, a one parameter degeneration to a TFT whose boundary conditions are given by classical Hamiltonian $G$-spaces, and which encodes a variant of the Hodge filtration on character varieties.

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We would like to acknowledge the National Science Foundation for its support through individual grants DMS-1103525 (DBZ) and DMS-1502178 (DN).

Received 2 February 2018

Accepted 9 September 2018

Published 27 November 2019