The Character Field Theory and Homology of Character Varieties

2017/05/11 by Ben-Zvi, David, Gunningham, Sam, Nadler, David · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1705.04266

Abstract

We construct an extended oriented (2+ε)-dimensional topological field theory, the character field theory 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 (N=4 d=4 super-Yang-Mills in the GL twist), and a universal version of the unipotent character field theory introduced in arXiv:0904.1247. Boundary conditions in 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 D(G) of D-modules on G. We show that the circle integral X_G(S¹) (the center and trace of D(G)) is identified with the category D(G/G) of "class D-modules", while for an oriented surface S (with arbitrary decorations at punctures) we show that X_G(S)≃\rm H_*^BM(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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