Klein Topological Field Theories from Group Representations

TL;DR

This paper demonstrates how finite group representations naturally produce open-closed and Klein topological field theories, linking algebraic properties of representations to topological quantum field theories.

Contribution

It establishes a direct connection between finite group representations and topological field theories, including the relation of correlators to Frobenius-Schur indicators and real division rings.

Findings

Finite group representations generate open-closed and Klein TFTs.

The 1-point correlator for the projective plane relates to Frobenius-Schur indicators.

Complex simple Klein TFTs correspond to real division rings.

Abstract

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.