Boundary minimal models and the Rogers-Ramanujan identities

TL;DR

The paper characterizes when irreducible modules L(c_{p,q}, h_{m,n}) over Vir_{p,q} are classically free by employing Li's filtration and jet algebras to pass from quantum to classical limits, reducing the problem to the structure of the Zhu C2-algebra and its jet algebra JR_V. It shows that classical freeness occurs exactly for boundary minimal models Vir_{2,2s+1}, with explicit descriptions of gr_F(L(c_{2,2s+1}, h_{1,i})) tied to partition ideals P^{s,i} and refined characters that mirror Andrews–Gordon Rogers–Ramanujan identities. The results provide a full description of the classical limits of these modules and interpret key combinatorial identities in terms of vertex Poisson algebras and PBW bases. For p,q>2, no irreducible Vir_{p,q}-module is classically free, underscoring a sharp boundary between boundary minimal models and the wider Virasoro landscape.

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 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.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Theorem 3
• Example 2.1: PBW filtration of $U(\mathfrak{g})$
• Proposition 2.2
• Example 2.3: $\mathop{\mathrm{gr}}\nolimits(M(c, h))$
• Example 2.4: $\mathop{\mathrm{ch}}\nolimits_{L(1/2, 0)}(t, q)$
• Proposition 3.1: nozaradan_introduction_2008
• Lemma 3.2: nozaradan_introduction_2008
• Lemma 3.3