Explicit classical limits of irreducible Virasoro minimal-model modules: closed Poisson generators for the boundary family and the associated graded beyond it

The simple Virasoro vertex operator algebras (minimal models) $\mathrm{Vir}_{p,q}$ have irreducible modules $L(c_{p,q},h_{m,n})$, and their classical limits — the associated graded modules $\mathrm{gr}_F(L)$ for the Poincare–Birkhoff–Witt-type filtration, which are modules over the vertex Poisson algebra $\mathrm{gr}_F(\mathrm{Vir}_{p,q})$ — control singular supports and feed the chiral-homology program. Salazar (J. Pure Appl. Algebra 230 (2026) 108281) determined exactly which of these modules are classically free: precisely the boundary minimal models $L(c_{2,2s+1},h_{1,i})$, for which he gave a complete presentation of the classical limit in terms of the Andrews–Gordon partition identities; the presentation writes the Poisson ideal through generators $u_1,\dots,u_i$ of which only $u_1=sL_{-2}^{s-1}L_{-1}$ is explicit, and for $p,q>2$ the modules fail classical freeness with the actual structure of $\mathrm{gr}_F$ left uncomputed. The open problem is to supply the missing explicit structure: closed-form descriptions of the generators $u_2,\dots,u_i$ of the Poisson submodule and of the module $R_{L(c_{2,2s+1},h_{1,i})}$ in the boundary family, and descriptions of the kernel of the canonical arc-space epimorphism $JR_{\mathrm{Vir}_{p,q}}\otimes_{R_{\mathrm{Vir}_{p,q}}}R_L\to\mathrm{gr}_F(L)$ for the non-boundary minimal models, of which the Ising case $(3,4)$ is the first test case. A full solution completes the representation-theoretic interpretation of the Andrews–Gordon identities and supplies module-level classical-limit data used in the study of singular supports and chiral homology.

Background

Virasoro minimal models and their modules. The Virasoro algebra $\mathrm{Vir}=\bigoplus_{n\in\mathbb{Z}}\mathbb{C}L_n\oplus\mathbb{C}C$ is the central extension of the Lie algebra of vector fields on the circle, with $[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}(m^3-m)C/12$. A vertex operator algebra (VOA) is the rigorous algebraic axiomatization of the chiral part of a two-dimensional conformal field theory. The universal Virasoro VOA $\mathrm{Vir}^c$ at central charge $c$ is generated by a single field of conformal weight $2$, and its simple quotient at $c_{p,q}=1-6(p-q)^2/pq$ (coprime $p,q\ge 2$) is the simple Virasoro VOA $\mathrm{Vir}_{p,q}$, a minimal model of conformal field theory. Its irreducible modules are $L(c_{p,q},h_{m,n})$ with $h_{m,n}=((np-mq)^2-(p-q)^2)/4pq$ and $0<m<p$, $0<n<q$. The subfamily $p=2$, $q=2s+1$ ($s\ge 1$) is called the boundary family (the Lee–Yang model $(2,5)$ is the case $s=2$; $s=1$ gives the trivial model). The maximal proper ideal of the vacuum module is generated by a singular vector $a_{p,q}$ — a highest-weight vector killed by all $L_n$, $n>0$ — of conformal weight $(p-1)(q-1)$; more generally, writing $M(c,h)$ for the Verma module of highest weight $h$ and $J(c,h)$ for its maximal proper submodule, one has $L(c,h)=M(c,h)/J(c,h)$.

Classical limits. Every Virasoro module carries the Poincare–Birkhoff–Witt (PBW) filtration $F$, on which a monomial $L_{-\lambda_1}\cdots L_{-\lambda_m}|v\rangle$ is weighted by the excess $\Delta(\lambda)-\mathrm{clen}(\lambda)$ of the total weight $\Delta(\lambda)=\lambda_1+\dots+\lambda_m$ over the conformal length $\mathrm{clen}(\lambda)$, which counts each part of $\lambda$ greater than $1$ as $2$ and each part equal to $1$ as $1$; the associated graded $\mathrm{gr}_F(M)$ is a module over the vertex Poisson algebra $\mathrm{gr}_F(\mathrm{Vir}^c)$, a commutative differential algebra with Poisson bracket $\{a,b\}=a_{(0)}b$ that is the quasi-classical (arc-space) limit of the VOA. Here $\sigma_0$ denotes the symbol map sending an element of a filtered module to its class in the associated graded. Two further commutative invariants are the Zhu $C_2$-algebra $R_V=V/C_2(V)$, where $C_2(V)$ is the span of the modes $a_{(-2)}b$, a Poisson algebra whose spectrum $X_V$ is the associated variety of $V$, and its module counterpart $R_M=M/C_2M$. The jet (arc) algebra $JR$ of a commutative algebra $R$ is obtained by freely adjoining all iterated derivatives $\partial^k a$. There is a canonical epimorphism $\phi_M:JR_V\otimes_{R_V}R_M\to\mathrm{gr}_F(M)$, and $M$ is called classically free when $\phi_M$ is an isomorphism, i.e. when the classical limit is the free arc module generated by $R_M$; at the algebra level, the singular support of $V$ is the closed subscheme of the arc space of $X_V$ whose ideal sheaf is the kernel of the corresponding epimorphism $\pi:JR_V\twoheadrightarrow\mathrm{gr}_F(V)$. Van Ekeren–Heluani proved that $\mathrm{Vir}_{p,q}$ itself is classically free if and only if $p=2$ — equivalently, $\mathrm{gr}_F(\mathrm{Vir}_{p,q})$ coincides with the arc algebra $JR_{\mathrm{Vir}_{p,q}}=\mathbb{C}[L_{-2},L_{-3},\dots]/(L_{-2}^{(p-1)(q-1)/2})_{\partial}$ exactly for $p=2$ and is a proper quotient of it otherwise — and classical freeness underlies their vanishing theorems for the first chiral homology of vertex algebras on elliptic curves (a vertex-algebraic analogue of de Rham homology on curves, relevant to finiteness of conformal blocks).

The source paper. Salazar determined classical freeness for all irreducible minimal-model modules: $L(c_{2,2s+1},h_{1,i})$ ($i=1,\dots,s$) is classically free, and no irreducible $\mathrm{Vir}_{p,q}$-module with $p,q>2$ is. For the boundary family the paper proves a refined-character formula (an Andrews–Gordon-type fermionic sum, i.e. a finite sum of $q$-series with products $\prod_j(1-q^j)^{-1}$ in the denominator), a PBW basis indexed by a set $P^{s,i}$ of partitions, and a presentation of the classical limit: $R_{L(c_{2,2s+1},h_{1,i})}\cong(\bigoplus_{k\in\mathbb{N}}\mathbb{C}[L_{-2}]L_{-1}^{k})/(L_{-2}^{s},u_1,\dots,u_i)_{\mathrm{psn}}$, where $(\cdot)_{\mathrm{psn}}$ denotes the Poisson submodule generated by the given elements, $u_k=\sigma_0((a_{2,2s+1})_{(k-1)}|c,h\rangle)$ for $k=1,\dots,i-1$ and $u_i=\sigma_0(w)$ with $w$ the singular vector of the module with leading term $L_{-1}^{i}|c,h\rangle$; the leading parts are $\ell p(u_k)=L_{-2}^{s-k}L_{-1}^{k}$ for $k=1,\dots,i-1$ and $\ell p(u_i)=L_{-1}^{i}$, and only $u_1=\{L_{-2}^{s},1\}=sL_{-2}^{s-1}L_{-1}$ has a closed form. The author remarks that he thinks there is no simple expression for $u_2,\dots,u_i$ and for $R_{L(c_{2,2s+1},h_{1,i})}$. The constructions are confined to $p=2$: for $p,q>2$ it is only shown that $\phi_M$ is not an isomorphism, and the kernel is not computed. For the Ising model $\mathrm{Vir}_{3,4}$ the VOA-level kernel is known (Andrews–van Ekeren–Heluani: the kernel of the epimorphism from the arc algebra $\mathbb{C}[L_{-2},L_{-3},\dots]/(L_{-2}^{3})_{\partial}$ onto $\mathrm{gr}_F(\mathrm{Vir}_{3,4})$ is the differential ideal generated by $b=L_{-4}L_{-3}L_{-2}+\tfrac{1}{6}L_{-5}L_{-2}^{2}$), and Salazar's earlier work produced explicit monomial bases and refined characters for an associated graded of the three irreducible Ising modules; but no Poisson-submodule presentation of the above type is known beyond the boundary family.

Partition combinatorics. A partition $\lambda=[\lambda_1\ge\lambda_2\ge\dots]$ of an integer is a nonincreasing finite sequence of positive integers (its parts); a partition ideal is a set of partitions closed under passing to subpartitions, and it has order $r$ if the minimal forbidden subpartitions have at most $r$ parts, so that order $1$ is a pure frequency condition. Two partition ideals are equivalent if they have the same generating function. Gordon's combinatorial generalization of the Rogers–Ramanujan identities, analytically extended to odd moduli by Andrews, asserts the equivalence $A^{s,i}\sim P^{s,i}$, where $A^{s,i}$ is the set of partitions with no part congruent to $0,\pm i$ modulo $2s+1$ (an order-$1$ ideal) and $P^{s,i}$ is the set of partitions avoiding a finite set $R^{s,i}$ of forbidden subpartitions (an order-$2$ ideal). Andrews' 1976 book already posed the problem of characterizing the equivalence classes of partition ideals; it is solved for order $1$ (Theorem 8.4 there) and open beyond. The source paper realizes $P^{s,i}$ as the index set of a PBW basis of $L(c_{2,2s+1},h_{1,i})$, which is the advertised natural interpretation of the Andrews–Gordon identities.

Problem Statement

Complete the explicit description of the classical limits of the irreducible Virasoro minimal-model modules $L(c_{p,q},h_{m,n})$, in the two linked directions left open by the characterization of classical freeness.

(a) Boundary family ($p=2$). For $s\ge 1$ and $i\in\{1,\dots,s\}$ (the question is non-vacuous for $i\ge 2$; for $i=1$ the presentation below is already explicit), determine the Poisson module $R_{L(c_{2,2s+1},h_{1,i})}\cong(\bigoplus_{k\in\mathbb{N}}\mathbb{C}[L_{-2}]L_{-1}^{k})/\sigma_0(J(c_{2,2s+1},h_{1,i}))$ explicitly: find closed-form expressions for the generators $u_2,\dots,u_i$ of the Poisson submodule $\sigma_0(J)=(L_{-2}^{s},u_1,\dots,u_i)_{\mathrm{psn}}$ — only $u_1=sL_{-2}^{s-1}L_{-1}$ is known, the $u_k$ being otherwise available only through their defining mode formulas $u_k=\sigma_0((a_{2,2s+1})_{(k-1)}|c,h\rangle)$ for $k<i$ and $u_i=\sigma_0(w)$, with leading parts $L_{-2}^{s-k}L_{-1}^{k}$ for $k<i$ and $L_{-1}^{i}$ for $u_i$ — or an equivalent closed presentation of the module or of $\mathrm{gr}_F(L(c_{2,2s+1},h_{1,i}))$. Completion standard for (a): the deliverable is resolved by any of the following, established for all $s\ge1$ and $i\in\{1,\dots,s\}$ — an explicit expression for each generator $u_k$ as a polynomial in the $L_{-n}$ with coefficients given by a closed formula in $s$ and $i$; a proven finite recursion or Gröbner-basis description that determines the Poisson submodule $\sigma_0(J)=(L_{-2}^{s},u_1,\dots,u_i)_{\mathrm{psn}}$; or an explicit differential-ideal presentation of $\mathrm{gr}_F(L(c_{2,2s+1},h_{1,i}))$ (equivalently an explicit presentation of the Poisson module $R_{L(c_{2,2s+1},h_{1,i})}$). Restating the defining mode formulas $u_k=\sigma_0((a_{2,2s+1})_{(k-1)}|c,h\rangle)$ together with the leading parts inherits only the information already recorded above and does not by itself resolve (a); an assertion that no such presentation exists does not resolve (a) either, since (a) asks for the explicit structure, not for a proof of its absence.

(b) Non-boundary family ($p,q>2$). For coprime $p,q>2$ and all admissible $m,n$, describe the associated graded module $\mathrm{gr}_F(L(c_{p,q},h_{m,n}))$ as a module over the vertex Poisson algebra $\mathrm{gr}_F(\mathrm{Vir}_{p,q})$, itself a proper quotient of the arc algebra $JR_{\mathrm{Vir}_{p,q}}=\mathbb{C}[L_{-2},L_{-3},\dots]/(L_{-2}^{(p-1)(q-1)/2})_{\partial}$: compute the kernel of the canonical epimorphism $\phi:JR_{\mathrm{Vir}_{p,q}}\otimes_{R_{\mathrm{Vir}_{p,q}}}R_{L}\to\mathrm{gr}_F(L)$, or give an otherwise closed presentation of $\mathrm{gr}_F(L)$. The Ising family $(p,q)=(3,4)$ — where the VOA-level kernel is the differential ideal generated by $L_{-4}L_{-3}L_{-2}+\tfrac{1}{6}L_{-5}L_{-2}^{2}$ and monomial bases of an associated graded of the three irreducible modules are known — is the first test case; answers must be consistent with these precedents and with the known characters of the modules.

References

1. Diego Salazar, Boundary minimal models and the Rogers-Ramanujan identities, Journal of Pure and Applied Algebra 230(6) (2026), article 108281, DOI 10.1016/j.jpaa.2026.108281, https://arxiv.org/abs/2405.07126
2. Diego Salazar, PBW bases of irreducible Ising modules, Journal of Algebra 639 (2024) 398-421, DOI 10.1016/j.jalgebra.2023.10.015, https://arxiv.org/abs/2306.07920
3. George E. Andrews, The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press (1998; reprint of the 1976 original), DOI 10.1017/CBO9780511608650, ISBN 9780521637664
4. Basil Gordon, A Combinatorial Generalization of the Rogers-Ramanujan Identities, American Journal of Mathematics 83(2) (1961) 393-396, DOI 10.2307/2372962, https://doi.org/10.2307/2372962
5. George E. Andrews, An Analytic Generalization of the Rogers-Ramanujan Identities for Odd Moduli, Proceedings of the National Academy of Sciences 71(10) (1974) 4082-4085, DOI 10.1073/pnas.71.10.4082, https://doi.org/10.1073/pnas.71.10.4082
6. Clemens Bruschek, Hussein Mourtada, Jan Schepers, Arc spaces and the Rogers–Ramanujan identities, The Ramanujan Journal 30(1) (2013) 9-38, DOI 10.1007/s11139-012-9401-y, https://doi.org/10.1007/s11139-012-9401-y
7. Jethro van Ekeren, Reimundo Heluani, Chiral Homology of Elliptic Curves and the Zhu Algebra, Communications in Mathematical Physics 386(1) (2021) 495-550, DOI 10.1007/s00220-021-04026-w, https://doi.org/10.1007/s00220-021-04026-w
8. George E. Andrews, Jethro van Ekeren, Reimundo Heluani, The Singular Support of the Ising Model, International Mathematics Research Notices 2023(10) (2023) 8800-8831, DOI 10.1093/imrn/rnab328, https://arxiv.org/abs/2005.10769
9. Shashank Kanade, Classical freeness of $\widehat{\mathfrak{sl}}_n$ at level 1 via combinatorics, arXiv:2606.19234 (2026), https://arxiv.org/abs/2606.19234
10. Bailin Song, Xianlong Zeng, Zhu's algebra and the $C_2$-algebra of a classically free vertex operator algebra, arXiv:2606.14407 (2026), https://arxiv.org/abs/2606.14407
11. A. Zuevsky, The first chiral homology group in higher genus, arXiv:2607.24719 (2026), https://arxiv.org/abs/2607.24719
12. Shane Chern, Zhitai Li, Linked partition ideals and Kanade–Russell conjectures, Discrete Mathematics 343 (2020) 111876, DOI 10.1016/j.disc.2020.111876, https://arxiv.org/abs/1809.08655
13. Shane Chern, Linked partition ideals and Russell’s three-colored partitions, arXiv:2607.03977 (2026), https://arxiv.org/abs/2607.03977
14. Haijun Li, Linked partition ideals and gap-frequency partitions, arXiv:2606.16739 (2026), https://arxiv.org/abs/2606.16739
15. Pooneh Afsharijoo, Looking for a New Version of Gordon's Identities, Annals of Combinatorics 25 (2021) 543-571, DOI 10.1007/s00026-021-00530-x, https://doi.org/10.1007/s00026-021-00530-x
16. Alapan Ghosh, Rupam Barman, J-generalization of the Rogers–Ramanujan–Gordon identities via commutative algebra, The Ramanujan Journal 70 (2026), DOI 10.1007/s11139-026-01394-6, https://doi.org/10.1007/s11139-026-01394-6