Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models

TL;DR

This paper explores the Kazhdan-Lusztig equivalence in logarithmic Virasoro models, explicitly describing modules of the associated quantum group and calculating their tensor products, leading to conjectured fusion rules.

Contribution

It provides an explicit description of indecomposable modules and their tensor products in the quantum group, and conjectures fusion rules for Virasoro Kac modules in LCFTs.

Findings

Explicit classification of quantum group modules

Calculation of tensor products of indecomposable modules

Conjectured fusion rules for Virasoro Kac modules

Abstract

The subject of our study is the Kazhdan-Lusztig (KL) equivalence in the context of a one-parameter family of logarithmic CFTs based on Virasoro symmetry with the (1,p) central charge. All finite-dimensional indecomposable modules of the KL-dual quantum group - the "full" Lusztig quantum sl(2) at the root of unity - are explicitly described. These are exhausted by projective modules and four series of modules that have a functorial correspondence with any quotient or a submodule of Feigin-Fuchs modules over the Virasoro algebra. Our main result includes calculation of tensor products of any pair of the indecomposable modules. Based on the Kazhdan-Lusztig equivalence between quantum groups and vertex-operator algebras, fusion rules of Kac modules over the Virasoro algebra in the (1,p) LCFT models are conjectured.