Frobenius--Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $\pi:\hat{G} \twoheadrightarrow \{1,-1\}$, a $\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B \hat{G}$ and a character $\lambda: \hat{G} \rightarrow U(1)$. The underlying oriented theory is a twisted Dijkgraaf-Witten theory. The construction is based in the $(\hat{G}, \hat{\theta},\lambda)$-twisted Real representation theory of $\ker \pi$. In particular, twisted Real representations are boundary conditions and the generalized Frobenius-Schur element is its crosscap state.