Structured Topological Field Theories via Crossed Simplicial Groups

TL;DR

This paper develops a classification framework for topological field theories using crossed simplicial groups, linking algebraic structures with geometric topological categories, and generalizing Frobenius algebras.

Contribution

It introduces a novel approach to classify topological field theories via crossed simplicial groups, connecting algebraic and topological structures in a new way.

Findings

Classifies topological field theories using crossed simplicial groups.

Shows equivalence to algebras with group actions and non-degenerate traces.

Generalizes the concept of Frobenius algebras.

Abstract

We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are equivalent to algebras equipped with a group action and a non-degenerate trace satisfying certain invariance requirements which generalize the notion of a frobenius algebra.