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On the resolution of categorical symmetries in (Non-) Unitary Rational CFTs

Arpan Bhattacharyya, Saptaswa Ghosh, Sounak Pal, Jagannath Santara

TL;DR

This work develops a general, SymTFT-free RCFT framework for symmetry-resolved entanglement entropy (SREE) governed by categorical symmetries. By expressing SREE directly through modular data and boundary-state constructions, it covers unitary and non-unitary, diagonal and non-diagonal RCFTs, including fusion multiplicities, and examines various boundary types (weakly/strongly symmetric and cloaking). The authors provide explicit calculations in Ising, M(6,5), SU(2)_{10} E6-type theories, the three-state Potts model, and several Haagerup–Izumi theories, uncovering when full sector resolution is possible and when it requires auxiliary boundary-defect data. The results illuminate how boundary conditions and fusion category structure shape entanglement distributions, offering a unified lens to study categorical symmetries in 2D critical systems and guiding extensions to non-unitary and non-diagonal contexts.

Abstract

We explore several aspects of the categorical symmetry-resolved entanglement entropy (SREE) in two-dimensional Rational Conformal Field Theories (RCFTs) and express it directly in terms of the modular data of the theory. Motivated by arXiv:2409.02806, we provide a general formula for SREE that applies to symmetric (weakly/strongly) and cloaking boundary conditions as well as for fusion rings with multiplicities without invoking any SymTFT construction, relying instead on a purely 2d RCFT analysis. We check the formula against several explicit examples. Additionally, we study symmetry resolution for both categorical and invertible symmetries in (non-)diagonal RCFTs and comment on the subtleties that arise in these cases. Finally, we extend our analysis to diagonal non-unitary RCFTs, focusing on theories with generalized Haagerup-Izumi modular data, and find full agreement with the given formula.

On the resolution of categorical symmetries in (Non-) Unitary Rational CFTs

TL;DR

This work develops a general, SymTFT-free RCFT framework for symmetry-resolved entanglement entropy (SREE) governed by categorical symmetries. By expressing SREE directly through modular data and boundary-state constructions, it covers unitary and non-unitary, diagonal and non-diagonal RCFTs, including fusion multiplicities, and examines various boundary types (weakly/strongly symmetric and cloaking). The authors provide explicit calculations in Ising, M(6,5), SU(2)_{10} E6-type theories, the three-state Potts model, and several Haagerup–Izumi theories, uncovering when full sector resolution is possible and when it requires auxiliary boundary-defect data. The results illuminate how boundary conditions and fusion category structure shape entanglement distributions, offering a unified lens to study categorical symmetries in 2D critical systems and guiding extensions to non-unitary and non-diagonal contexts.

Abstract

We explore several aspects of the categorical symmetry-resolved entanglement entropy (SREE) in two-dimensional Rational Conformal Field Theories (RCFTs) and express it directly in terms of the modular data of the theory. Motivated by arXiv:2409.02806, we provide a general formula for SREE that applies to symmetric (weakly/strongly) and cloaking boundary conditions as well as for fusion rings with multiplicities without invoking any SymTFT construction, relying instead on a purely 2d RCFT analysis. We check the formula against several explicit examples. Additionally, we study symmetry resolution for both categorical and invertible symmetries in (non-)diagonal RCFTs and comment on the subtleties that arise in these cases. Finally, we extend our analysis to diagonal non-unitary RCFTs, focusing on theories with generalized Haagerup-Izumi modular data, and find full agreement with the given formula.

Paper Structure

This paper contains 19 sections, 125 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Fusion category and tube algebra.
  3. Categorical symmetry–resolved entanglement.
  4. Non-unitarity and symmetry resolution.
  5. Goal and brief takeaway.
  6. Background and basic setup
  7. Cloaking boundary.
  8. Symmetry resolved entanglement entropy (SREE)
  9. Computing SREE in Unitary Rational CFT
  10. Symmetry resolution with respect to Tambara-Yamagami (TY) Fusion Category for Ising CFT
  11. Symmetry resolution for $\mathcal{M}(6,5)$
  12. Symmetry resolution in non-diagonal RCFTs
  13. SREE in Non-Unitary Rational CFT
  14. Lee-Yang Minimal Model $\mathcal{M}(5, 2)$: a simplest example
  15. The Haagerup Rational CFT $\mathcal{R}_{\kappa=3}$
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Inserting a topological defect line($\mathcal{L}$) along the Euclidean time direction (left), which, after the conformal transformation (from cylinder to plane), acts on the local operator (right).
  • Figure 2: For a set of intervals A and B, as the Hilbert space is not a direct product, we have put conformal boundary conditions with two small $\epsilon$ circles at the entangling surface, which turn into two different Cardy boundary states after conformal mapping to an annulus.