Non-unitary fusion categories and their doubles via endomorphisms

TL;DR

This paper constructs non-unitary fusion categories using subfactor methods, computes their doubles and modular data, and conjectures simple forms for their S and T matrices, with implications for conformal field theories.

Contribution

It generalizes the Haagerup-Izumi family to non-unitary cases and provides explicit endomorphism realizations of models like Yang-Lee.

Findings

Constructed non-unitary fusion categories via subfactor methods.

Computed quantum doubles and modular data for these categories.

Conjectured simple forms for the modular S and T matrices.

Abstract

We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup-Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang-Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman-Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matrices of the doubles of these fusion categories. We would expect all of these doubles to be realised as the category of modules of a rational VOA and conformal net of factors. We expect our approach will also suffice to realise the non-semisimple tensor categories arising in logarithmic conformal field theories.