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

TL;DR

This paper develops a new two-dimensional unoriented topological field theory using twisted Real representation theory of finite graded groups, extending Dijkgraaf-Witten theory with Frobenius-Schur indicators.

Contribution

It introduces a novel construction of unoriented topological field theories based on twisted Real representations and Frobenius-Schur indicators for finite graded groups.

Findings

Constructed a 2D unoriented open/closed topological field theory from finite graded groups.

Linked twisted Real representation theory to boundary conditions and crosscap states.

Extended Dijkgraaf-Witten theory with new algebraic structures.

Abstract

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.