Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gauging Noninvertible Defects: A 2-Categorical Perspective

Thibault D. Décoppet, Matthew Yu

TL;DR

The paper extends the notion of anomaly to noninvertible surface-symmetries by formulating condensation in multifusion 2-categories. It develops a comprehensive framework for condensing separable algebras in braided, sylleptic, and symmetric 2-categories, producing new (multi)fusion 2-categories and identifying when a fibre 2-functor to $2\mathbf{Vec}$ or $2\mathbf{SVec}$ exists. A key contribution is the classification of strongly fusion theories via higher cohomology: braided cases are governed by SH$^5$ and ordinary cohomology, while symmetric cases admit a 2-Deligne-type theorem yielding a fibre functor to $2\mathbf{SVec}$; obstructions are traced to specific cohomology groups (e.g., $H^7(E[4];\mathbb{C}^\times)$ in the bosonic symmetric setting). The work provides explicit examples (totally disconnected vs connected, bosonic vs fermionic) and lays out cohomological obstructions to condensation, highlighting the physical interpretation in terms of gauging higher-form and noninvertible symmetries. Overall, the results advance a robust topological understanding of noninvertible symmetries in higher categories and their potential role in theories without global symmetries.

Abstract

We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surface operators using the framework of condensation in 2-categories. Given a multifusion 2-category, potentially with some additional levels of monoidality, we prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. We give examples where the resulting category displays grouplike fusion rules and through a cohomology computation, find the obstruction to condensing further to the vacuum theory.

Gauging Noninvertible Defects: A 2-Categorical Perspective

TL;DR

The paper extends the notion of anomaly to noninvertible surface-symmetries by formulating condensation in multifusion 2-categories. It develops a comprehensive framework for condensing separable algebras in braided, sylleptic, and symmetric 2-categories, producing new (multi)fusion 2-categories and identifying when a fibre 2-functor to or exists. A key contribution is the classification of strongly fusion theories via higher cohomology: braided cases are governed by SH and ordinary cohomology, while symmetric cases admit a 2-Deligne-type theorem yielding a fibre functor to ; obstructions are traced to specific cohomology groups (e.g., in the bosonic symmetric setting). The work provides explicit examples (totally disconnected vs connected, bosonic vs fermionic) and lays out cohomological obstructions to condensation, highlighting the physical interpretation in terms of gauging higher-form and noninvertible symmetries. Overall, the results advance a robust topological understanding of noninvertible symmetries in higher categories and their potential role in theories without global symmetries.

Abstract

We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surface operators using the framework of condensation in 2-categories. Given a multifusion 2-category, potentially with some additional levels of monoidality, we prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. We give examples where the resulting category displays grouplike fusion rules and through a cohomology computation, find the obstruction to condensing further to the vacuum theory.
Paper Structure (29 sections, 29 theorems, 39 equations, 21 figures)

This paper contains 29 sections, 29 theorems, 39 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on 2-Categories
  3. Graphical Calculus
  4. Braided Monoidal 2-Categories
  5. Sylleptic and Symmetric Monoidal 2-Categories
  6. Algebras and Modules
  7. Higher Condensations and Separable Algebras
  8. Relative Tensor Product
  9. Braided and Symmetric Algebras
  10. Definitions
  11. The 2-Category of Modules over a Braided Algebra.
  12. The 2-Category of Modules over a Symmetric Algebra
  13. Specific 2-Category of Modules
  14. Totally Disconnected 2-Category
  15. Bosonic case
  16. ...and 14 more sections

Key Result

Theorem 1

For $\mathfrak{B}$ a braided multifusion 2-category, condensing a braided separable algebra $B$ in $\mathfrak{B}$ results in a multifusion 2-category.

Figures (21)

  • Figure 1: This gives a three dimensional view of condensing the algebra $A$, taking place in a 2-category $\mathfrak{C}$. The resulting boundary is the category $\mathbf{Mod}_{\mathfrak{C}}(A)$, and $2\mathbf{SVec}$ represents the "fermionic vacuum".
  • Figure 2: Axiom a (Part 1)
  • Figure 3: Axiom a (Part 2)
  • Figure 4: Axiom a (Part 3)
  • Figure 5: Axiom a (Part 4)
  • ...and 16 more figures

Theorems & Definitions (70)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 60 more