Boundary minimal models and the Rogers-Ramanujan identities

We determine when the irreducible modules $L(c_{p,q},h_{m,n})$ over the simple Virasoro vertex algebras ${\mathrm{Vir}}_{p,q}$, 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 ${\mathrm{Vir}}_{2,2s+1}$ 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.