Existence of simple rational vertex operator algebras over positive-characteristic fields with more than one simple module

Vertex operator algebras (VOAs) are traditionally studied over the complex numbers, where rational VOAs with several simple modules, such as the Virasoro minimal models, are the basic building blocks of rational conformal field theory. Over fields of positive characteristic the landscape changes drastically: the Heisenberg VOA ceases to be simple and its simple quotients have a unique simple module, and Griffin (2026) proved that the simple Virasoro VOAs $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ are rational and holomorphic for all central charges in the prime field when $p=3$ or $5$, and for $p=7$ with $c\notin\{3,6\}$. The only proposed example of a simple rational VOA in positive characteristic with more than one simple module, Dong and Ren's Ising-model VOA $L(1/2,0)_{\mathbb{F}}$ (2016), was shown by Griffin to be incorrectly analyzed in characteristics 3, 5, and 7, where the VOA is in fact holomorphic. This record states the resulting open problem: whether any VOA over a positive-characteristic field can be simple, rational, and possess more than one simple module up to isomorphism (Question 5.30 of Griffin), together with its principal concrete instance, the conjecture that $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ is rational and holomorphic for every field of characteristic $p>2$ and every $c\in\mathbb{F}$ (Conjecture 5.29).

Background

A vertex operator algebra (VOA) is the algebraic axiomatization of the chiral algebra of a two-dimensional conformal field theory: an $\mathbb{N}$-graded vector space $V=\bigoplus_{n\geq 0}V_n$ with finite-dimensional graded pieces, a vacuum vector, and a state-field map $v\mapsto Y(v,z)=\sum_{k\in\mathbb{Z}}v_k z^{-k-1}$ whose modes satisfy locality and the Jacobi identity, together with a distinguished conformal vector $\omega\in V_2$. The modes of $\omega$ furnish a representation of the Virasoro algebra, the unique nontrivial central extension of the Lie algebra of polynomial vector fields on the punctured line, with brackets $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$; the scalar $c$ is the central charge, and $L_0$ recovers the grading. Although VOAs are usually studied over $\mathbb{C}$, the axioms make sense over any commutative ring and in particular over any field $\mathbb{F}$, and the general representation-theoretic framework over arbitrary fields is due to Dong and Ren.

A weak $V$-module is a vector space with compatible vertex-operator actions; it is admissible if it carries an $\mathbb{N}$-grading $W=\bigoplus_{s\geq 0}W(s)$ with finite-dimensional homogeneous components such that a mode $v_k$ of a homogeneous element $v$ of degree $d$ maps $W(s)$ into $W(s+d-k-1)$. A VOA is rational if every admissible module is a direct sum of simple admissible modules, and holomorphic if it is rational and the adjoint module $V$ itself is, up to isomorphism, the only simple admissible module. The main computational tool is the Zhu algebra $A(V)$, an associative algebra built as a quotient of $V$, whose simple modules are in bijection with the simple admissible $V$-modules; in particular, a holomorphic VOA has, up to isomorphism, a single simple $A(V)$-module (for the Virasoro quotients in Griffin's Theorem 5.28 one in fact gets $A(L_{\mathrm{Vir}}(c,0)_{\mathbb{F}})\cong\mathbb{F}$). A VOA is $C_2$-cofinite if $V/C_2(V)$ is finite-dimensional, where $C_2(V)$ is spanned by the modes $u_{-2}v$. Over $\mathbb{C}$, a fundamental recent criterion of McRae states that an $\mathbb{N}$-graded, simple, self-contragredient (isomorphic to its contragredient dual), $C_2$-cofinite VOA with semisimple Zhu algebra is rational; over $\mathbb{C}$, rational $C_2$-cofinite VOAs also underpin semisimple modular tensor categories and the theory of conformal blocks, where spaces of coinvariants form finite-rank vector bundles on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable $n$-pointed curves — reproducing this geometric picture over positive-characteristic fields is a major goal articulated in the source paper.

For a field $\mathbb{F}$ and $c\in\mathbb{F}$, the universal Virasoro VOA $V_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ is freely generated by a conformal vector of central charge $c$, and $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ denotes its simple quotient. Over $\mathbb{C}$, Wang proved that the discrete-series (minimal model) Virasoro VOAs, with central charges $c_{r,s}=1-6(r-s)^2/(rs)$ for coprime integers $r,s>1$, are rational; Griffin extends this to all fields of characteristic 0 via integral forms and Zhu algebra computations. The case $c_{3,4}=1/2$ is the Ising minimal model $L(1/2,0)$, rational over $\mathbb{C}$ with exactly three simple modules of conformal weights ($L_0$-eigenvalues) $0$, $1/2$, and $1/16$.

In positive characteristic the familiar objects behave differently. The Heisenberg (free boson) VOAs are no longer simple over prime characteristic, and Li and Mu showed that their simple quotients are rational with a unique simple module, so the free boson provides no example of a rational VOA with several modules. Jiao, Li, and Mu studied the restricted Virasoro VOA $V^0(c,0)_{\mathbb{F}}$ (a quotient of the universal Virasoro VOA tied to the $p$-center of the Virasoro algebra): its Zhu algebra is $\mathbb{F}[x]/(x^p-x)$, which is semisimple with $p$ one-dimensional simple modules, and it is $C_2$-cofinite; yet Griffin proved that $V^0(c,0)_{\mathbb{F}}$ is not rational for any $c$ — so in positive characteristic a semisimple Zhu algebra no longer guarantees rationality, and McRae's characteristic-zero criterion has no known positive-characteristic analogue. Griffin's proof of the small-characteristics holomorphicity theorem rests on explicit computation: the Shapovalov form, the contravariant bilinear form on a highest-weight Virasoro module determined by $(L_{-n}v,w)=(v,L_nw)$, has a Gram matrix on each graded component whose null space detects singular vectors (highest-weight vectors), and these null spaces determine the image of the maximal ideal in the Zhu algebra.

Against this background, Dong and Ren (2016) had proposed the first and still only candidate example: they claimed that $L(1/2,0)_{\mathbb{F}}$ is rational with exactly three simple modules over any field of characteristic not equal to $2$ or $7$ (two simple modules in characteristic 7, with rationality unproven there), using a free-fermion realization, and they conjectured in general that reducing an integral form of a rational VOA yields simple rational VOAs for all but finitely many primes. Griffin's computations show that these structure claims are incorrect in characteristics 3, 5, and 7, where $L(1/2,0)_{\mathbb{F}}$ is holomorphic; the status of the claim in characteristics $p\geq 11$ is unverified. Most recently, lattice vertex algebras over positive-characteristic fields have been studied as another candidate family: $V_{L,\mathbb{F}}$ is simple exactly when $p\nmid\det L$ (Griess and Lam treat the degenerate case $p\mid\det L$, showing the radical of the invariant form is the unique maximal ideal), and Mu and Zhao proved that for a simply laced ADE root lattice with $p\nmid\det L$, $V_{L,\mathbb{F}}$ is isomorphic to the level-one simple affine vertex algebra and classified its irreducible $\mathbb{N}$-graded modules — but rationality in positive characteristic has not been established for any of these algebras.

Problem Statement

Let $\mathbb{F}$ be a field of positive characteristic. Recall that a VOA over $\mathbb{F}$ is rational if every admissible module is a direct sum of simple admissible modules, and holomorphic if it is rational and the adjoint module is, up to isomorphism, its only simple module. The problem is the question posed as Question 5.30 of Griffin (arXiv:2607.05584):

Does there exist a vertex operator algebra $V$ over a field $\mathbb{F}$ of positive characteristic that is simple, rational, and has more than one simple module up to isomorphism? Equivalently: must every simple rational VOA over a positive-characteristic field be holomorphic?

The principal concrete instance, posed by the source as Conjecture 5.29, concerns the Virasoro family: for a field $\mathbb{F}$ of characteristic $p>2$ and $c\in\mathbb{F}$, let $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ be the simple quotient of the universal Virasoro VOA of central charge $c$. Is $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ rational with the adjoint module as its only simple module for every $p>2$ and every $c\in\mathbb{F}$? This is proven for $c\in\mathbb{F}_p$ when $p=3$ or $p=5$, and for $p=7$ with $c\in\mathbb{F}_7\setminus\{3,6\}$ (Griffin, Theorem 5.28); the cases $p=7$, $c\in\{3,6\}$, all $p\geq 11$, and all $c\in\mathbb{F}\setminus\mathbb{F}_p$ are open. The only prior claimed example, Dong and Ren's $L(1/2,0)_{\mathbb{F}}$ for characteristic not in $\{2,7\}$, is known to be incorrectly analyzed in characteristics 3, 5, and 7, where the VOA is in fact holomorphic; its status in characteristic at least 11 is unverified. A Virasoro VOA $L_{\mathrm{Vir}}(c,0)_{\mathbb{F}}$ that is rational with a second simple module would answer the main question positively, while a proof of the conjecture would eliminate the most natural candidate family; lattice vertex algebras $V_{L,\mathbb{F}}$ with $p\nmid\det L$ and level-one affine vertex algebras are the other natural candidates.

The question as posed places no restriction on the characteristic; note that the frameworks in the cited literature exclude characteristic 2 for the Virasoro family.

References

1. Colton Griffin, Modular Zhu algebra theory and Virasoro vertex algebras, arXiv:2607.05584 (2026). https://arxiv.org/abs/2607.05584
2. Chongying Dong and Li Ren, Vertex operator algebras associated to the Virasoro algebra over an arbitrary field, Transactions of the American Mathematical Society 368 (2016) 5177-5196, doi:10.1090/tran/6529, arXiv:1308.0087. https://arxiv.org/abs/1308.0087
3. Chongying Dong and Li Ren, Representations of vertex operator algebras over an arbitrary field, Journal of Algebra 403 (2014) 497-516, doi:10.1016/j.jalgebra.2014.01.007. https://doi.org/10.1016/j.jalgebra.2014.01.007
4. Haisheng Li and Qiang Mu, Heisenberg VOAs over fields of prime characteristic and their representations, Transactions of the American Mathematical Society 370 (2017) 1159-1184, doi:10.1090/tran/7094, arXiv:1501.04314. https://doi.org/10.1090/tran/7094
5. Xiangyu Jiao, Haisheng Li and Qiang Mu, Modular Virasoro vertex algebras and affine vertex algebras, Journal of Algebra 519 (2019) 273-311, doi:10.1016/j.jalgebra.2018.10.029, arXiv:1711.00985. https://doi.org/10.1016/j.jalgebra.2018.10.029
6. Weiqiang Wang, Rationality of Virasoro vertex operator algebras, International Mathematics Research Notices 1993 (7) (1993) 197-211, doi:10.1155/S1073792893000212. https://doi.org/10.1155/S1073792893000212
7. Robert McRae, On rationality for $C_2$-cofinite vertex operator algebras, Cambridge Journal of Mathematics 14 (1) (2026) 1-115, doi:10.4310/cjm.260225023743, arXiv:2108.01898. https://doi.org/10.4310/cjm.260225023743
8. Robert L. Griess and Ching Hung Lam, Lattice vertex algebras over a field of positive characteristic: degenerate cases, arXiv:2607.28933 (2026). https://arxiv.org/abs/2607.28933
9. Qiang Mu and Hongju Zhao, Lattice vertex algebras of type ADE over fields of prime characteristic and their representations, arXiv:2608.20706 (2026). https://arxiv.org/abs/2608.20706