On the $(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2$ fusion category

TL;DR

The paper analyzes non-invertible categorical symmetries in two-dimensional CFTs by computing the modular data for thefusion category $(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2$. It develops the tube algebra framework, constructs irreducible representations, and derives the 22-by-22 modular S matrix (and diagonal T matrix) that govern partition-function transforms in the presence of non-invertible symmetry lines. The authors provide a pedagogical, self-contained treatment of the direct and semidirect product constructions, the lasso intertwiners, and the Drinfeld center, with explicit results for the Fib-based semidirect product, including detailed sector decompositions across untwisted and twisted Hilbert spaces. This work establishes the essential representation-theoretic data needed for a modular conformal bootstrap analysis of putative non-rational Virasoro CFTs with this large category of symmetry, and highlights the distinctive features due to non-invertibility. The explicit 22-sector structure and the associated partition-function basis pave the way for future numerical bootstrap investigations and comparisons with concrete CFT constructions.

Abstract

There might exist non-rational Virasoro CFTs in two dimensions with a $(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2$ 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.

Figures (1)

• Figure 1: Junction $i \times j \to k$.