Fusion rules and logarithmic representations of a WZW model at fractional level

TL;DR

This paper analyzes the fusion rules and representation structures of the su(2) WZW model at fractional level -4/3, revealing non-bounded spectra and non-diagonalisable actions of L_0, and classifies the fusion-closed representations.

Contribution

It provides a detailed analysis of fusion products at fractional level, identifying non-semisimple representations and deriving the complete fusion rules for the model.

Findings

Fusion products include representations with unbounded L_0 spectrum.

Some fusion products are not completely reducible.

The complete set of fusion-closed representations is identified.

Abstract

The fusion products of admissible representations of the su(2) WZW model at the fractional level k=-4/3 are analysed. It is found that some fusion products define representations for which the spectrum of L_0 is not bounded from below. Furthermore, the fusion products generate representations that are not completely reducible and for which the action of L_0 is not diagonalisable. The complete set of representations that is closed under fusion is identified, and the corresponding fusion rules are derived.