On the Higher Categorical Structure of Topological Defects in Quantum Field Theories

TL;DR

This paper develops a comprehensive mathematical framework using higher category theory to describe topological defects in quantum field theories with various tangential structures, unifying and extending previous results.

Contribution

It introduces a structured higher categorical approach to topological defects, generalizing known models and assuming the stratified cobordism hypothesis for stable structures.

Findings

Unified description of topological defects using higher dagger categories

Recovers known results for oriented defects in 2D QFTs

Proves the framework under the stratified cobordism hypothesis

Abstract

We propose a unifying mathematical framework describing the higher categorical structures formed by topological defects in quantum field theory equipped with tangential structures, such as orientations, framings, or $\operatorname{Pin}^{\pm}$-structures, in terms of structured versions of higher dagger categories. This recovers all previously known results, including the description of oriented topological defects in 2-dimensional quantum field theories by pivotal bicategories. Assuming the stratified cobordism hypothesis, we prove our proposal for topological defects with stable tangential structures that admit a direct sum in fully extended topological quantum field theories.