  In this paper, we introduce Topological Quantum Field Theories (TQFTs)
generalizing the arithmetic computations done by Hausel and
Rodr\'iguez-Villegas and the geometric construction done by Logares, Mu\~noz,
and Newstead to study cohomological invariants of $G$-representation varieties
and $G$-character stacks. We show that these TQFTs are related via a natural
transformation that we call the 'arithmetic-geometric correspondence'
generalizing the classical formula of Frobenius on the irreducible characters
of a finite group. We use this correspondence to extract some information on
the character table of finite groups using the geometric TQFT, and vice versa,
we greatly simplify the geometric calculations in the case of upper triangular
matrices by lifting its irreducible characters to the geometric setting.
