The singular support of the Virasoro minimal models beyond the Ising model: finite generation and explicit structure of the kernel differential ideal

To every vertex algebra $V$ one attaches its singular support, the closed subscheme of the arc space of its associated scheme cut out by the kernel $\ker\pi$ of the canonical surjection from the arc algebra $JR_V$ of the Zhu $C_2$-algebra onto the Li-associated graded $\mathrm{gr}_F V$. Andrews, van Ekeren and Heluani proved that for the Ising model $\mathrm{Vir}_{3,4}$ this kernel is, as a differential ideal, principal — generated by a single element $b$ of degree 9 — making the Ising model the first known non-classically-free vertex algebra with a finitely generated singular-support ideal; the same computation yielded a monomial basis, a Gröbner basis, and a new fermionic character formula that is modular as a linear combination of Nahm sums for the matrix $\begin{pmatrix}8&3\\3&2\end{pmatrix}$. For every higher model $\mathrm{Vir}_{3,p'}$ they showed that $\ker\pi$ contains a distinguished element $b^{(p')}$ of degree $2p'+1$, but that for $p'\ge 5$ the induced surjection $JR_V/(b^{(p')})_\partial\twoheadrightarrow\mathrm{gr}_F V$ is not injective — already in degree 19 when $p'=5$ — leaving the structure of $\ker\pi$ unknown. The problem is to describe the singular support of the Virasoro minimal models beyond the Ising model, beginning with $\mathrm{Vir}_{3,5}$, and to settle the question, first posed by Arakawa and Linshaw for $W_3$ and repeated by Andrews, van Ekeren and Heluani, of whether $\ker\pi$ is finitely generated as a differential ideal for other non-classically-free rational ($C_2$-cofinite) vertex algebras: explicit generators, Gröbner bases and leading-term ideals, together with the induced monomial bases, refined characters, and Rogers–Ramanujan-type partition identities. The core problem described above remains open.

Background

A vertex operator algebra (VOA) is the algebraic structure underlying a two-dimensional chiral conformal field theory: a vector space of fields carrying a meromorphic operator product expansion, strongly generated by finitely many modes. The Virasoro minimal models $\mathrm{Vir}_{p,p'}$, for coprime integers $2\le p<p'$, are the simple quotients of the universal Virasoro vertex algebra of central charge $c_{p,p'}=1-6(p'-p)^2/(pp')$; they are rational (finitely many irreducible modules) and $C_2$-cofinite (the quotient algebra $R_V$ below is finite-dimensional), and include the Ising model $\mathrm{Vir}_{3,4}$ ($c=1/2$).

Two geometric invariants are attached to any VOA $V$. The Li filtration $F^\bullet V$ is a decreasing filtration compatible with the vertex-algebra operations, and its associated graded $\mathrm{gr}_F V$ is a commutative Poisson vertex algebra — a commutative algebra with a derivation $\partial$ and a Poisson bracket of degree $-1$ — viewed as the classical limit of $V$. Its degree-zero part is Zhu's $C_2$-algebra $R_V=V/C_2(V)$, the quotient by the subspace spanned by the products $a_{-2}b$ of fields, whose spectrum $X_V$ is the associated scheme of $V$; for a minimal model $R_V\cong\mathbb{C}[L]/(L^{(p-1)(p'-1)/2})$, a fat point supported at the origin (a single point carrying nilpotent functions). The arc space $J_\infty X_V$ is the scheme of formal arcs on $X_V$; its coordinate ring $JR_V$ is the differential algebra $\mathbb{C}[L_{-2},L_{-3},\dots]/(L_{-2}^{(p-1)(p'-1)/2})_\partial$ with derivation $\partial L_{-n}=(n-1)L_{-n-1}$, and the differential ideal $(\dots)_\partial$ generated by a set is the ordinary ideal generated by the set together with all of its $\partial$-derivatives. Arc-space coordinate rings are not Noetherian as differential algebras, and differential ideals in them need not be finitely generated.

There is a canonical surjection of Poisson vertex algebras $\pi\colon JR_V\twoheadrightarrow\mathrm{gr}_F V$. If $\pi$ is an isomorphism, $V$ is called classically free. The singular support $SS(V)=\mathrm{Spec}\,\mathrm{gr}_F V$ is in general a closed subscheme of the arc space $J_\infty X_V$, cut out by the differential ideal $\ker\pi$. Arakawa and Linshaw, who introduced this circle of ideas, proved the closed embedding $SS(V)\hookrightarrow J_\infty X_V$ and asked — in the setting of the simple $W_3$ algebra at central charge $-2$ (the extension of the Virasoro algebra by one additional primary field of conformal weight 3), whose associated scheme is the cusp $\mathrm{Spec}\,\mathbb{C}[\ell,w]/(w^2-\ell^3)$ — whether the differential ideal describing the singular support is finitely generated; in their worked example this ideal is the nilradical of the arc algebra, which they expected to be generated as a differential ideal by two explicit nilpotent elements. Bourqui and Sebag had shown that a naturally arising differential ideal (the radical of $(XY)$ in a two-variable differential polynomial ring) is not differentially finitely generated, so finite generation is genuinely subtle. van Ekeren and Heluani proved that $\mathrm{Vir}_{p,p'}$ is classically free if and only if $p=2$ (the boundary models $\mathrm{Vir}_{2,2s+1}$, where the statement amounts to a vertex-algebraic incarnation of the Andrews–Gordon identities, the classical two-parameter generalization of the Rogers–Ramanujan product-sum partition identities), and showed that the classical-freeness condition controls chiral homology — the cohomology theory of a vertex algebra coupled to the geometry of an algebraic curve: when $SS(V)\cong J_\infty X_V$, the degree-one chiral homology of an elliptic curve degenerating to the nodal curve specializes to the Hochschild homology of Zhu's algebra (a second associative algebra attached to $V$, built from its zero modes, whose modules are those of $V$; not to be confused with the $C_2$-algebra $R_V$ above). Hence every minimal model with $p\ge3$ has $\ker\pi\neq0$, and the structure of this ideal became a central open question of the subject.

Andrews, van Ekeren and Heluani settled the first case. For the Ising model they proved $\ker\pi=(b)_\partial$ with $b=\frac16L_{-5}L_{-2}^2+L_{-4}L_{-3}L_{-2}$ of degree 9 — the first example of a non-classically-free vertex algebra whose singular-support ideal is finitely generated as a differential ideal. Their method produced an explicit Gröbner basis — a generating set whose leading monomials determine ideal membership and Hilbert series; here an infinite structured family of generators plus four exceptional elements — an explicit monomial basis of the Ising model in Virasoro modes, and a new fermionic (quasiparticle) formula for the Ising character: a linear combination of Nahm sums — $q$-series of the form $\sum_k q^{(1/2)k^TAk+k^TB+C}/\prod_i (q)_{k_i}$ with $(q)_k=\prod_{j=1}^k(1-q^j)$ — for the matrix $A=\begin{pmatrix}8&3\\3&2\end{pmatrix}$, remarkable because no single Nahm sum for this matrix was known to be modular despite computer searches motivated by Nahm's conjecture. Since the monomial basis is graded both by total degree and by the number of factors (the Li level), the same computation refines the character to a two-variable generating series. For the entire $(3,p')$ series they proved that $\ker\pi$ always contains a distinguished element $b^{(p')}=\frac{9-2p'}{3(p'-2)}L_{-5}L_{-2}^{p'-2}+L_{-4}L_{-3}L_{-2}^{p'-3}$ of degree $2p'+1$, giving a surjection $JR_V/(b^{(p')})_\partial\twoheadrightarrow\mathrm{gr}_F V$; but for $p'\ge5$ this morphism is not injective — already in degree 19 when $p'=5$ — so $\ker\pi$ is strictly larger than $(b^{(p')})_\partial$ and its structure is unknown. This is the first untreated case.

Problem Statement

Determine the structure of the singular support of the Virasoro minimal models beyond the Ising model, and of non-classically-free rational vertex algebras generally. The prototype is the Ising computation of Andrews, van Ekeren and Heluani; the questions, in order of concreteness, are the following; each is an independent open problem, so a complete answer to any one of them by itself resolves the task:

(1) First open case. For $V=\mathrm{Vir}_{3,5}$ (central charge $-3/5$), so that $R_V=\mathbb{C}[L]/(L^4)$ and $JR_V=\mathbb{C}[L_{-2},L_{-3},\dots]/(L_{-2}^4)_\partial$ with $\partial L_{-n}=(n-1)L_{-n-1}$: describe $\ker\pi\subset JR_V$ explicitly. Is $\ker\pi$ finitely generated as a differential ideal? If so, give an explicit finite generating set and describe the leading-term ideal $\mathrm{LT}(\ker\pi)$ — the ideal of leading monomials with respect to a declared monomial order (for instance grevlex, the graded reverse lexicographic order) — in exact analogy with the Ising case where $\ker\pi=(b)_\partial$ is principal. It is known that $\ker\pi$ strictly contains the principal differential ideal $(b^{(5)})_\partial$ generated by the degree-11 element $b^{(5)}=-\frac19L_{-5}L_{-2}^3+L_{-4}L_{-3}L_{-2}^2$, with kernel elements outside it first appearing in degree 19.

(2) The whole series. Answer the same questions for $\mathrm{Vir}_{3,p'}$ with $p'\ge5$ and for the general minimal models $\mathrm{Vir}_{p,p'}$ with $p\ge3$. The following deliverables are independent, and a complete answer to any one of them by itself resolves this question: (a) finite generation of $\ker\pi$; (b) explicit generators and the leading-term ideal; (c) — through the mechanism established in the Ising case, namely the equality of the Hilbert series of the quotient arc algebra with the minimal-model character — the induced monomial and PBW (Poincaré–Birkhoff–Witt-type) bases, the two-variable refined characters obtained from the bigrading by total degree and Li level, and Rogers–Ramanujan-type $q$-series identities for these models and their irreducible modules.

(3) The general phenomenon. Decide, for other non-classically-free rational ($C_2$-cofinite) vertex algebras — including the $W_3$ algebras (among them the simple $W_3$ algebra at $c=-2$, the setting of Arakawa and Linshaw's original question) — whether $\ker\pi$ is finitely generated as a differential ideal; or exhibit a vertex algebra in this class for which it is not, noting that differential ideals of arc-space rings need not be differentially finitely generated in general.

The common objective uniting these questions is the explicit, finitely presented description of the singular support as a differential subscheme of the arc space of the associated scheme, of which the Ising computation is the prototype and $\mathrm{Vir}_{3,5}$ the first untreated case.

References

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