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 $G = \text{SU}(2), \text{SO}(3)$ and $\text{U}(2)$. 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.