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Journal of High Energy Physics

On the (Fib ⊠ Fib) ⋊ S2 fusion category

Maddalena Ferragatta, Balt C. van Rees

Published Jun 16, 2026• 0 citations
Publisher

Abstract

A bstract There might exist non-rational Virasoro CFTs in two dimensions with a (Fib ⊠ Fib)⋊ S 2 categorical symmetry. We calculate the necessary ingredients for a modular conformal bootstrap analysis of these theories. After reviewing the basics of fusion categories, we present the irreducible representations, the lasso maps that intertwine between different Hilbert spaces, and finally the 22-by-22 modular S matrix. We highlight the peculiarities introduced by the non-invertible nature of the symmetry. This paper is written in a pedagogical manner and can therefore serve as an accessible entry point into the literature.

DOI
10.1007/jhep06(2026)174

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On the (Fib⊠Fib)⋊S2(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2(Fib⊠Fib)⋊S2​ fusion category
There might exist non-rational Virasoro CFTs in two dimensions with a (Fib⊠Fib)⋊S2(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2(Fib⊠Fib)⋊S2​ categorical symmetry. We calculate the necessary ingredients for a modular conformal bootstrap analysis of these theories. After reviewing the basics of fusion categories, we present the irreducible representations, the lasso maps that intertwine between different Hilbert spaces, and finally the 22-by-22 modular S matrix. We highlight the peculiarities introduced by the non-invertible nature of the symmetry. This paper is written in a pedagogical manner and can therefore serve as an accessible entry point into the literature.
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A bstract We initiate the S-matrix bootstrap analysis of theories with non-invertible symmetries in (1+1) dimensions. Our previous work [1] showed that crossing symmetry of S-matrices in such theories is modified, with modification characterized by the fusion category data. By imposing unitarity, symmetry and the modified crossing, we constrain the space of consistent S-matrices, identifying integrable theories with non-invertible symmetries at the cusps of allowed regions. We also extend the modified crossing rules to cases where vacua transform in non-regular representations of fusion category, utilizing a connection to a dual category CM∗ {\mathcal{C}}_{\mathcal{M}}^{\ast } CM∗​ C M ∗ and Symmetry Topological Field Theory (SymTFT). This highlights the utility of SymTFT in the analysis of scattering amplitudes.
We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c=1c = 1c=1 CFTs and uncover a rich structure. We find that even those c=1c = 1c=1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.
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Using the numerical modular bootstrap, we constrain the space of 1+1d1+1\mathrm{d}1+1d conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category C\mathcal{C}C. We derive universal and rigorous upper bounds on the lightest C\mathcal{C}C-preserving scalar local operator for fusion categories such as the Ising and Fibonacci categories. These numerical bounds constrain the possible robust gapless phases protected by a noninvertible global symmetry, which commonly arise from microscopic lattice models such as the anyonic chains. We also derive bounds on the lightest C\mathcal{C}C-violating local operator. Our bootstrap equations naturally arise from a ``slab construction,'' where the CFT is coupled to the 2+1d2+1\mathrm{d}2+1d Turaev-Viro topological quantum field theory , also known as the symmetry TFT.