Comments on nonunitary conformal field theories

TL;DR

This paper explores the properties of nonunitary rational conformal field theories, introduces the Galois shuffle relationship between primary fields, and classifies an infinite family of nonunitary WSU(3) models.

Contribution

It introduces the Galois shuffle relationship in nonunitary RCFTs and classifies a new family of nonunitary WSU(3) minimal models.

Findings

Identification of a primary field with minimal weight and positive S column.

Explanation of the multiplicity of the primary field o.

Classification of an infinite family of nonunitary WSU(3) minimal models.

Abstract

As is well-known, nonunitary RCFTs are distinguished from unitary ones in a number of ways, two of which are that the vacuum 0 doesn't have minimal conformal weight, and that the vacuum column of the modular S matrix isn't positive. However there is another primary field, call it o, which has minimal weight and has positive S column. We find that often there is a precise and useful relationship, which we call the Galois shuffle, between primary o and the vacuum; among other things this can explain why (like the vacuum) its multiplicity in the full RCFT should be 1. As examples we consider the minimal WSU(N) models. We conclude with some comments on fractional level admissible representations of affine algebras. As an immediate consequence of our analysis, we get the classification of an infinite family of nonunitary WSU(3) minimal models in the bulk.