Boundary minimal models and the Rogers-Ramanujan identities

TL;DR

This paper characterizes when irreducible modules over Virasoro vertex algebras are classically free, linking these cases to boundary minimal models and interpreting the Andrews-Gordon identities within this framework.

Contribution

It provides a complete classification of classical limits of Virasoro modules over boundary minimal models using the Rogers-Ramanujan identities.

Findings

Classical freeness occurs only for boundary minimal models $ ext{Vir}_{2, 2s+1}$.

Complete description of classical limits via jet algebra of Zhu $C_2$-algebra.

Provides a natural interpretation of Andrews-Gordon identities.

Abstract

We determine when the irreducible modules $L(c_{p, q}, h_{m, n})$ over the simple Virasoro vertex algebras $\operatorname{Vir}_{p, q}$, where $p, q \ge 2$ are relatively prime with $0 < m < p$ and $0 < n < q$, are classically free. It turns out that this only happens with the boundary minimal models, i.e., with the irreducible modules over $\operatorname{Vir}_{2, 2s + 1}$ for $s \in \mathbb{Z}_+$. We thus obtain a complete description of the classical limits of these modules in terms of the jet algebra of the corresponding Zhu $C_2$-algebra. The Andrews-Gordon generalization of the Rogers-Ramanujan identities is used in the proof, and our results in turn provide a natural interpretation of these identities.