Fusion Algebras of Logarithmic Minimal Models

TL;DR

This paper conjectures explicit fusion rules for logarithmic minimal models LM(p,p'), revealing their algebraic structure, indecomposable representations, and confirming these rules through numerical lattice studies, advancing understanding of logarithmic conformal field theories.

Contribution

It provides the first detailed conjecture of fusion rules for LM(p,p') models with explicit algebraic and numerical support, including indecomposable representations and their ranks.

Findings

Fusion rules are quasi-rational, involving reducible yet indecomposable representations.

Confirmed the presence of indecomposable representations of rank 3 in specific models.

Closure of the fusion algebra occurs without higher-rank indecomposables.

Abstract

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.