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Translation invariant defects as an extension of topological symmetries

Federico Ambrosino, Ingo Runkel, Gérard M. T. Watts

TL;DR

This work extends the conventional topological symmetry of 2d QFT by introducing translation invariant line defects, which admit non-singular fusion and form a monoidal category that contains topological defects as a subcategory. It provides a perturbative description via a chiral TFT off a bulk perturbation, establishing a commutation condition that determines when a defect remains translation invariant to all orders; translation invariant defects, under RG flow, flow to IR topological defects, enriching the detectable symmetry content. The authors construct the perturbative defect category $\mathcal{T}_{pert}$ with a precise fusion rule and derive functional relations in Ising and Lee-Yang models, constraining RG endpoints. They culminate with a categorical formulation in terms of Yetter-Drinfeld modules, showing how translation invariant defects correspond to Hopf-algebraic data and revealing non-semisimple, infinitely rich defect theories in minimal models. Overall, the paper provides both a perturbative mechanism and a rigorous algebraic framework for studying translation invariant defects as an extension of topological symmetries in two-dimensional QFT.

Abstract

The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings we focus on line defects in 2d QFT and we point out that topological defects naturally embed into a larger class, namely translation invariant defects. The latter still allow for non-singular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee-Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.

Translation invariant defects as an extension of topological symmetries

TL;DR

This work extends the conventional topological symmetry of 2d QFT by introducing translation invariant line defects, which admit non-singular fusion and form a monoidal category that contains topological defects as a subcategory. It provides a perturbative description via a chiral TFT off a bulk perturbation, establishing a commutation condition that determines when a defect remains translation invariant to all orders; translation invariant defects, under RG flow, flow to IR topological defects, enriching the detectable symmetry content. The authors construct the perturbative defect category with a precise fusion rule and derive functional relations in Ising and Lee-Yang models, constraining RG endpoints. They culminate with a categorical formulation in terms of Yetter-Drinfeld modules, showing how translation invariant defects correspond to Hopf-algebraic data and revealing non-semisimple, infinitely rich defect theories in minimal models. Overall, the paper provides both a perturbative mechanism and a rigorous algebraic framework for studying translation invariant defects as an extension of topological symmetries in two-dimensional QFT.

Abstract

The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings we focus on line defects in 2d QFT and we point out that topological defects naturally embed into a larger class, namely translation invariant defects. The latter still allow for non-singular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee-Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.

Paper Structure

This paper contains 5 sections, 2 theorems, 24 equations, 3 figures.

Table of Contents

  1. Topological line defects in two-dimensional field theory
  2. Translation invariant line defects
  3. The commutation condition
  4. Fusion of translation invariant defects and functional relations
  5. Mathematical description and Yetter-Drinfeld modules

Key Result

Proposition 5.1

Suppose all categories and functors are $\mathbb{C}$-linear. Then for all $\mu \in \mathbb{C}$, $\mu \neq 0$ we have $\mathcal{M}_{\omega} \cong \mathcal{M}_{\mu \omega}$ as additive monoidal categories.

Figures (3)

  • Figure 1: Topological defects along a renormalisation group flow. Only some of the topological defects (gray box) of the UV CFT are preserved under the perturbation by the bulk field $\varphi$. These flow to some of the topological defects of the IR CFT.
  • Figure 2: a) A patch of the CFT world sheet $\Sigma$ with an insertion of a bulk field $\varphi$. b) The same patch in the SymTFT representation on the three-manifold $\Sigma \times [0,1]$. c) Chiral TFT representation on the three-manifold $\Sigma \times [-1,1]$.
  • Figure 3: Translation invariant defects along a renormalisation group flow. The translation invariant defects compatible with the bulk perturbation flow to topological defects of the IR CFT.

Theorems & Definitions (4)

  • Definition 5.1
  • Proposition 5.1
  • proof
  • Proposition 5.2