Classification of Kac representations in the logarithmic minimal models LM(1,p)

TL;DR

This paper classifies Kac representations in logarithmic minimal models using algebraic and lattice methods, proposing a fusion algebra structure and testing it with computational algorithms.

Contribution

It introduces a classification of Kac representations as submodules of Feigin-Fuchs modules and conjectures their fusion algebra, extending it with contragredient modules.

Findings

Proposed a classification of Kac representations as submodules of Feigin-Fuchs modules.

Conjectured the structure of the Kac fusion algebra and its extension.

Validated the fusion algebra using lattice approaches and the Nahm-Gaberdiel-Kausch algorithm.

Abstract

For each pair of positive integers r,s, there is a so-called Kac representation (r,s) associated with a Yang-Baxter integrable boundary condition in the lattice approach to the logarithmic minimal model LM(1,p). We propose a classification of these representations as finitely-generated submodules of Feigin-Fuchs modules, and present a conjecture for their fusion algebra which we call the Kac fusion algebra. The proposals are tested using a combination of the lattice approach and applications of the Nahm-Gaberdiel-Kausch algorithm. We also discuss how the fusion algebra may be extended by inclusion of the modules contragredient to the Kac representations, and determine polynomial fusion rings isomorphic to the conjectured Kac fusion algebra and its contragredient extension.