Cohomology of character stacks via TQFTs

We study the cohomology of G-representation varieties and G-character stacks by means of a topological quantum field theory (TQFT). This TQFT is constructed as the composite of a so-called field theory and the 6-functor formalism of sheaves on topological stacks. We apply this framework to compute the cohomology of various G-representation varieties and G-character stacks of closed surfaces for G = SU(2), SO(3) and U(2). This work can be seen as a categorification of earlier work, in which such a TQFT was constructed on the level of Grothendieck groups to compute the corresponding Euler characteristics.