Orbifolds of n-dimensional defect TQFTs

TL;DR

This paper develops a general framework for n-dimensional defect TQFTs, introduces a method for constructing orbifold theories via algebraic data, and specializes the theory to the case of three dimensions.

Contribution

It defines defect TQFTs in arbitrary dimensions, introduces a general orbifold construction applicable to any n-dimensional defect TQFT, and connects to known models like state sums and symmetry gauging.

Findings

Established a universal orbifold construction for defect TQFTs

Demonstrated invariance under Pachner moves constrains algebraic data

Specialized the theory to three-dimensional TQFTs

Abstract

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincar\'e duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.