Extended Homotopy Quantum Field Theories and their Orbifoldization

TL;DR

This paper defines extended homotopy quantum field theories, develops an orbifold construction for G-equivariant theories, and explores their modular structures, providing a geometric and algebraic unification with broad generalizations.

Contribution

It introduces a bicategorical orbifold construction for extended equivariant topological field theories, unifying and generalizing algebraic orbifold concepts.

Findings

Orbifold construction relates equivariant theories to non-equivariant ones.

Analysis of modular structures in 3-2-1-dimensional theories.

Generalization of orbifoldization to pushforward operations along group morphisms.

Abstract

We present a precise definition of extended homotopy quantum field theories and develop an orbifold construction for these theories when the target space is the classifying space of a finite group $G$, i.e. for $G$-equivariant topological field theories. More precisely, we use a bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. Thereby, we give a unification, geometric underpinning and vast generalization of algebraic concepts of orbifoldization. In the special case of 3-2-1-dimensional equivariant topological field theories, we investigate the equivariant modular structure on the categories that such theories yield upon evaluation on the circle. By means of our orbifold construction this equivariant modular structure will be related to the modular structure on the orbifold category. We also generalize our orbifold construction to a pushforward operation along an arbitrary morphism of finite groups and hence provide a valuable tool for the construction of extended homotopy quantum field theories.