The quantum group dual of the first-row subcategory for the generic Virasoro VOA

TL;DR

This paper establishes a duality between a subcategory of modules of the generic Virasoro VOA and the representations of a quantum group, providing new proofs and insights into their structure and fusion rules.

Contribution

It develops a concrete duality between the first-row module subcategory of the Virasoro VOA and quantum group representations, with new proofs of fusion rules and associativity.

Findings

Proved the analyticity of intertwining operator compositions.

Demonstrated that conformal blocks are determined by quantum group methods.

Established the associativity governed by quantum group 6j-symbols.

Abstract

In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the $6j$-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$.