Boundary algebras and Kac modules for logarithmic minimal models

TL;DR

This paper develops an algebraic framework for understanding Virasoro Kac modules in logarithmic minimal models, linking lattice transfer operators with module structures and fusion rules, and providing explicit examples and character formulas.

Contribution

It introduces a quotient of the one-boundary Temperley-Lieb algebra to identify Virasoro Kac modules as submodules of Feigin-Fuchs modules, clarifying their structure and fusion properties.

Findings

Virasoro Kac modules are finitely generated submodules of Feigin-Fuchs modules.

The algebraic framework accurately reproduces fusion rules at the character and module levels.

Explicit examples confirm the module structures and fusion behavior.

Abstract

Virasoro Kac modules were initially introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.