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

Lukas Müller

TL;DR

This work develops a unifying framework to encode the higher-categorical structure of topological defects in quantum field theories with stratified tangential data. By introducing ${\mathcal B}_n$-dagger categories, defined as lax-limits of adjoint-categorical data over a pointed stratified tangential structure ${\mathcal B}_n$, the authors connect defect categories $\mathcal{D}^{\mathcal{B}^n}$ to structured higher daggers, recovering known results such as the equivalence between oriented 2D defects and pivotal bicategories. They propose concrete algebraic constructions—condensation/orbifold completions and Euler completions—within this framework, providing universal properties that govern gauging of generalized symmetries and the enrichment of pivotal/dagger structures. Under the stratified cobordism hypothesis, these results yield a principled path to model fully extended QFTs with tangential data, enabling systematic treatment of non-invertible defects and their gaugings across dimensions. Overall, the paper offers a cohesive, scalable apparatus to unify tangential-structure effects on defect theories and to derive new categorical invariants via completion procedures.

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.

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

TL;DR

This work develops a unifying framework to encode the higher-categorical structure of topological defects in quantum field theories with stratified tangential data. By introducing -dagger categories, defined as lax-limits of adjoint-categorical data over a pointed stratified tangential structure , the authors connect defect categories to structured higher daggers, recovering known results such as the equivalence between oriented 2D defects and pivotal bicategories. They propose concrete algebraic constructions—condensation/orbifold completions and Euler completions—within this framework, providing universal properties that govern gauging of generalized symmetries and the enrichment of pivotal/dagger structures. Under the stratified cobordism hypothesis, these results yield a principled path to model fully extended QFTs with tangential data, enabling systematic treatment of non-invertible defects and their gaugings across dimensions. Overall, the paper offers a cohesive, scalable apparatus to unify tangential-structure effects on defect theories and to derive new categorical invariants via completion procedures.

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 -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.
Paper Structure (7 sections, 3 theorems, 16 equations, 1 figure)

This paper contains 7 sections, 3 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tangential structures for stratified manifolds
  3. Structured higher dagger categories
  4. Categories of topological defects are higher dagger categories
  5. Algebraic constructions for categories of topological defects
  6. Condensation, orbifold, and unitary idempotent completions
  7. Euler completions

Key Result

Proposition 3.14

There is a functor ${\mathsf{RigidCat}}_{\bullet}\colon {\mathsf{Vect}}_{\operatorname{inj}}^{\operatorname{op}}\longrightarrow \operatorname{\mathscr{C}at}$ sending $\mathbb{R}^n$ to ${\mathsf{RigidCat}}_n$ equipped with the trivial $O_n$-action and the cobordism hypothesis action on morphisms.

Figures (1)

  • Figure 1: The value of the gauge theory on a 2-dimensional manifold $M$ constructed by the evaluation of the original theory in the presence of a network of defects.

Theorems & Definitions (27)

  • Example 2.3
  • Definition 2.5
  • Remark 2.6
  • Example 2.8
  • Example 2.10
  • Conjecture 3.1
  • Remark 3.2
  • Definition 3.4
  • Remark 3.5
  • Example 3.6
  • ...and 17 more