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

TL;DR

This paper explores the potential existence of non-rational Virasoro conformal field theories with a specific categorical symmetry, providing foundational calculations for future bootstrap analysis and highlighting unique features of non-invertible symmetries.

Contribution

It computes the irreducible representations, lasso maps, and the modular S matrix for the $( ext{Fib} oxtimes ext{Fib}) times S_2$ fusion category, facilitating bootstrap studies of related CFTs.

Findings

Calculated the irreducible representations of the fusion category

Derived the 22-by-22 modular S matrix

Identified peculiarities of non-invertible symmetries

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.