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Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT

Pavel Putrov, Rajath Radhakrishnan

TL;DR

The paper develops a symmetry-based framework to classify braidings of topological operators, linking the anomaly of non-invertible higher-form symmetries to braiding data captured by the SymTFT. By embedding a fusion category into its Drinfeld center and analyzing the bulk-to-boundary maps, the authors provide an algorithm to enumerate all braidings from the fusion data without solving the hexagon equations. They extend the method to 3+1d, showing braidings of fusion 2-categories occur only when the 3+1d SymTFT is a Dijkgraaf–Witten theory and derive necessary and sufficient conditions for several important 2-categories (ΣC, 2Vec$_G^{\pi}$, TY$(A,\pi)$) to admit braidings, with explicit results for abelian and Tambara–Yamagami cases. The work furnishes concrete, computable criteria based on modular data and Lagrangian algebras, enabling systematic classification of higher-form symmetry anomalies and their mixed anomalies across dimensions. This approach clarifies how invertible and non-invertible symmetries intertwine via condensation defects and paves the way for higher-dimensional generalizations and practical computations in QFT and TQFT settings.

Abstract

The anomaly of non-invertible higher-form symmetries is determined by the braiding of topological operators implementing them. In this paper, we study a method to classify braidings on topological line and surface operators by leveraging the fact that topological operators which admit a braiding are symmetries of their associated SymTFT. This perspective allows us to formulate an algorithm to explicitly compute all possible braidings on a given fusion category, bypassing the need to solve the hexagon equations. Additionally, using 3+1d SymTFTs, we determine braidings on various fusion 2-categories. We prove a necessary and sufficient condition for the fusion 2-categories $Σ\mathcal{C}$, 2Vec$_G^π$ and Tambara-Yamagami (TY) 2-categories TY$(A,π)$ to admit a braiding.

Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT

TL;DR

The paper develops a symmetry-based framework to classify braidings of topological operators, linking the anomaly of non-invertible higher-form symmetries to braiding data captured by the SymTFT. By embedding a fusion category into its Drinfeld center and analyzing the bulk-to-boundary maps, the authors provide an algorithm to enumerate all braidings from the fusion data without solving the hexagon equations. They extend the method to 3+1d, showing braidings of fusion 2-categories occur only when the 3+1d SymTFT is a Dijkgraaf–Witten theory and derive necessary and sufficient conditions for several important 2-categories (ΣC, 2Vec, TY) to admit braidings, with explicit results for abelian and Tambara–Yamagami cases. The work furnishes concrete, computable criteria based on modular data and Lagrangian algebras, enabling systematic classification of higher-form symmetry anomalies and their mixed anomalies across dimensions. This approach clarifies how invertible and non-invertible symmetries intertwine via condensation defects and paves the way for higher-dimensional generalizations and practical computations in QFT and TQFT settings.

Abstract

The anomaly of non-invertible higher-form symmetries is determined by the braiding of topological operators implementing them. In this paper, we study a method to classify braidings on topological line and surface operators by leveraging the fact that topological operators which admit a braiding are symmetries of their associated SymTFT. This perspective allows us to formulate an algorithm to explicitly compute all possible braidings on a given fusion category, bypassing the need to solve the hexagon equations. Additionally, using 3+1d SymTFTs, we determine braidings on various fusion 2-categories. We prove a necessary and sufficient condition for the fusion 2-categories , 2Vec and Tambara-Yamagami (TY) 2-categories TY to admit a braiding.

Paper Structure

This paper contains 34 sections, 9 theorems, 157 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Braidings on topological lines operators from 2+1d SymTFT
  3. 2+1d SymTFT
  4. Computing the modular data of $\mathcal{Z}(\mathcal{C})$
  5. Classifying braidings on topological line operators
  6. Examples
  7. Braidings on $\text{Vec}_G^{\omega}$
  8. Braidings on Rep$(G)$
  9. $G=\mathds{Z}_2$
  10. $G=\mathds{Z}_2 \times \mathds{Z}_2$
  11. $G=S_3$
  12. Braiding on fusion categories admitting a modular braiding
  13. Braidings on Tambara-Yamagami fusion category
  14. Braidings on topological surfaces and lines from 3+1d SymTFT
  15. SymTFT for Invertible 0-form Symmetry
  16. ...and 19 more sections

Key Result

Theorem 3.1

Vec$_G^{\omega}$ admits a braiding if and only if $G$ is abelian and $\tau_g(\omega)=d\psi_g$ for some 2-cochain $\psi_g\in C^2(G,U(1))$ for all $g\in G$.

Figures (10)

  • Figure 1: Double braiding between two line operators is captured by the $S$-matrix.
  • Figure 2: Braiding the line $a$ with $b$ through the bulk using the maps $\iota$ and $F$.
  • Figure 3: Lasso action corresponding to the tube algebra basis element $\mathcal{T}_{abc}^d$.
  • Figure 4: The fusion of $([g],\pi) \in \mathcal{D}$ on the gapped boundary $\mathcal{B}_{\text{2Vec}_G^{\omega}}$ must produce the line operator $g\in \text{Vec}_G^{\omega}$. The junction vector space $V_{([g],\pi_g),g}$ is 1-dimensional with a basis vector $i$.
  • Figure 5: Composition of tube algebra action on the boundary line $g$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Theorem 3.1
  • Proposition 3.1
  • Proposition 4.1
  • Proposition 5.1
  • Theorem 5.1
  • Corollary 5.1
  • Corollary 5.2
  • Theorem 5.2
  • Corollary 5.3