Homotopy field theory in dimension 2 and group-algebras

TL;DR

This paper develops a framework for homotopy quantum field theories in dimension 2, classifies them using crossed group-algebras, and introduces state sum models and a Verlinde formula variant.

Contribution

It introduces cohomological (1+1)-dimensional HQFTs, classifies them via crossed group-algebras, and constructs new state sum models.

Findings

Classified (1+1)-dimensional HQFTs using crossed group-algebras.

Developed two state sum models for (1+1)-dimensional HQFTs.

Proved that these HQFTs are sums of rescaled cohomological HQFTs.

Abstract

We apply the idea of a topological quantum field theory (TQFT) to maps from manifolds into topological spaces. This leads to a notion of a (d+1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed d-dimensional manifolds and (d+1)-dimensional cobordisms endowed with homotopy classes of maps into a given space. For a group $\pi$, we introduce cohomological HQFT's with target $K(\pi,1)$ derived from cohomology classes of $\pi$ and its subgroups of finite index. The main body of the paper is concerned with (1+1)-dimensional HQFT's. We classify them in terms of so called crossed group-algebras. In particular, the cohomological (1+1)-dimensional HQFT's over a field of characteristic 0 are classified by simple crossed group-algebras. We introduce two state sum models for (1+1)-dimensional HQFT's and prove that the resulting HQFT's are direct sums of rescaled cohomological HQFT's. We also discuss a version of the Verlinde formula in this setting.