A boundary characterization of Turaev-Viro TQFTs

TL;DR

This paper characterizes boundary conditions in 3D TQFTs, providing explicit locality conditions that enable state sum constructions, and demonstrates their applicability to Turaev-Viro and Dijkgraaf-Witten theories.

Contribution

It introduces explicit boundary locality conditions for 3D TQFTs and proves their satisfaction in Turaev-Viro and Dijkgraaf-Witten models, enabling state sum formulations.

Findings

Turaev-Viro models obey boundary locality conditions

Dijkgraaf-Witten theories with boundary defects satisfy these conditions

Boundary locality enables state sum constructions for these theories

Abstract

We consider three-dimensional topological field theories on manifolds with boundary defects and identify explicit boundary locality conditions. These conditions imply a state sum construction of the given TQFT. As a consistency check, we prove that Turaev-Viro state sum models obey the boundary locality conditions. Recent progress (arXiv:2410.18049v1) in the description of defects in Dijkgraaf-Witten theories enables us to show that these theories likewise satisfy boundary locality. This directly implies that Dijkgraaf-Witten theories with boundary defects admit a state sum description.