A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence

TL;DR

This paper proves a conjectural equivalence between categories of representations of small quantum groups and certain vertex operator algebras using algebraic methods, contingent on analytic properties of the vertex algebra.

Contribution

It provides a conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence, extending the understanding of the relationship between quantum groups and vertex algebras.

Findings

Established an algebraic proof of the conjectural equivalence.

Characterized the quantum group category via accessible algebraic quantities.

Relied on analytic assumptions about the vertex algebra's representation category.

Abstract

The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such an equivalence, and more general versions, using mainly algebraic arguments that characterize the representation category of the quantum group by quantities that are accessible on the vertex algebra side. Our proof is conditional on suitable analytic properties of the vertex algebra and its representation category. More precisely, we assume that it is a finite braided rigid monoidal category where the Frobenius-Perron dimensions are given by asymptotics of analytic characters.