Fusion of irreducible modules in WLM(p,p')

TL;DR

This paper constructs and analyzes a fusion algebra for W-extended logarithmic minimal models, revealing new indecomposable modules and symmetries, and connecting to polynomial fusion rings.

Contribution

It introduces a comprehensive fusion algebra including reducible indecomposable modules and relates it to known algorithms and polynomial rings.

Findings

Fusion algebra includes reducible indecomposable modules.

The algebra reproduces known results for specific models.

Additional modules are introduced to ensure invariance under conjugation.

Abstract

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p,p'). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM(1,p') and WLM(2,3). For p>1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces an additional (p-1)(p'-1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights Delta_{kp-1,1}, k=1,2,3. We also identify polynomial fusion rings associated with our fusion algebras.