Polynomial fusion rings of W-extended logarithmic minimal models

TL;DR

This paper constructs and analyzes polynomial fusion rings for W-extended logarithmic minimal models, providing explicit fusion matrices and addressing the missing identity issue for p>1.

Contribution

It introduces a fusion-matrix realization of the fusion algebra, identifies isomorphic fusion rings, and proposes extensions to incorporate the identity element.

Findings

Fusion algebra closes without identity for p>1

Explicit fusion matrices are constructed

Extensions and quotients of fusion rings are analyzed

Abstract

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p') can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W. Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1. Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it. We also consider various extensions of it and quotients thereof, and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings. One particular extension is reminiscent of the fundamental fusion algebra of LM(p,p') and offers a natural way of introducing the missing identity for p>1. Working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual Moore-Penrose inverses intertwining between the W-fundamental and enlarged fusion algebras.