Mathematical Research Letters

Volume 26 (2019)

Number 5

The character field theory and homology of character varieties

Pages: 1313 – 1342



David Ben-Zvi (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Sam Gunningham (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

David Nadler (Department of Mathematics, University of California at Berkeley)


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.

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