Orbifolds of $n$–dimensional defect 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 Poincaré 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 $n=3$.