Fusion Rules of the ${\cal W}_{p,q}$ Triplet Models

TL;DR

This paper determines the fusion rules and constructs the Grothendieck group for the logarithmic ${ m W}_{p,q}$ triplet models, enabling the derivation of projective covers and a candidate modular invariant partition function.

Contribution

It generalizes the fusion rules and modular invariants from the ${ m W}_{2,3}$ model to arbitrary ${ m W}_{p,q}$ triplet models, providing a comprehensive framework.

Findings

Fusion rules for ${ m W}_{p,q}$ models are explicitly determined.

A Grothendieck group with consistent product structure is constructed.

A candidate modular invariant partition function is proposed.

Abstract

In this paper we determine the fusion rules of the logarithmic ${\calW}_{p,q}$ triplet theory and construct the Grothendieck group with subgroups for which consistent product structures can be defined. The fusion rules are then used to determine projective covers. This allows us also to write down a candidate for a modular invariant partition function. Our results demonstrate that recent work on the ${\cal W}_{2,3}$ model generalises naturally to arbitrary (p,q).