Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry

Clay Cordova, Diego García-Sepúlveda, Nicholas Holfester

TL;DR

This work develops a unified framework for understanding how non-invertible fusion-category symmetries constrain the massive particle spectra in (1+1)d QFTs. By linking bulk fusion data to boundary module categories and analyzing open-sector maps, it derives a general procedure to predict symmetry-enforced degeneracies that do not rely on integrability. Applying the method to RG flows from minimal models, it reproduces known degeneracy patterns—threefold multiplets in the Fibonacci case and fourfold/2(m−2) multiplets in Tambara-Yamagami and related flows—solely from topological symmetry data and IR TQFT structure. The results offer a robust, RG-invariant approach to classify particle multiplets in theories with non-invertible symmetries and provide a practical toolkit for analyzing flows in minimal-model dynamics.

Abstract

We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry

TL;DR

This work develops a unified framework for understanding how non-invertible fusion-category symmetries constrain the massive particle spectra in (1+1)d QFTs. By linking bulk fusion data to boundary module categories and analyzing open-sector maps, it derives a general procedure to predict symmetry-enforced degeneracies that do not rely on integrability. Applying the method to RG flows from minimal models, it reproduces known degeneracy patterns—threefold multiplets in the Fibonacci case and fourfold/2(m−2) multiplets in Tambara-Yamagami and related flows—solely from topological symmetry data and IR TQFT structure. The results offer a robust, RG-invariant approach to classify particle multiplets in theories with non-invertible symmetries and provide a practical toolkit for analyzing flows in minimal-model dynamics.

Abstract

We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.
Paper Structure (21 sections, 2 theorems, 114 equations, 2 figures, 1 table)

This paper contains 21 sections, 2 theorems, 114 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fusion Categories, Module Categories, and Degeneracies
  3. Relevant Background
  4. Fusion Categories (Bulk)
  5. Module Categories (Boundary)
  6. Open Sectors
  7. Degeneracies
  8. Compositions
  9. Kernels and Cokernels
  10. Particle Degeneracies in Massive QFTs
  11. $\mathcal{C}$-Symmetric TQFTs
  12. Lagrangian Algebras
  13. General Procedure
  14. Particle-Soliton Degeneracies in RG Flows from Minimal Models
  15. Tricritical Ising Deformed by the $\phi_{2,1}$ Operator
  16. ...and 6 more sections

Key Result

Theorem 1

Take $\mathcal{M}$ a (left) $\mathcal{C}$-module category for $\mathcal{C}$ a fusion category. There exists an algebra $\mathcal{A}$ in $\mathcal{C}$ such that $\mathcal{M} \simeq \mathcal{C}_{\mathcal{A}}$ with $\mathcal{C}_{\mathcal{A}}$ the category of (right) $\mathcal{A}$ modules in $\mathcal{C

Figures (2)

  • Figure 1: Schematic depiction of the associated Landau-Ginzburg potential of the Tricritical Ising model deformed by the $\phi_{2,1}$ primary operator. Note in particular the absence of a $\mathbb{Z}_{2}$ symmetry in the potential.
  • Figure 2: Schematic depiction of the Landau-Ginzburg potential corresponding to the negative-sign deformation of the Tricritical Ising model deformed by the $\phi_{1,3}$ primary operator. Note in particular the presence of three vacua, in spite of the fact that the only internal group-like symmetry present along the RG flow is $\mathbb{Z}_{2}$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2