Zhu's conjecture: finite-dimensionality of the $C_2$-algebra of every rational vertex operator algebra

Zhu's conjecture, originating in Zhu's 1996 work on modular invariance, asserts that every $\mathbb{N}$-graded rational vertex operator algebra (VOA) $V$ has a finite-dimensional $C_2$-algebra $R_V=V/C_2(V)$, i.e. is $C_2$-cofinite. Song and Zeng (2026) proved the conjecture for classically free VOAs by showing that then $R_V\cong\operatorname{gr}A(V)$, the associated graded of Zhu's algebra, and simultaneously proved that non-trivial holomorphic VOAs of CFT type — including $L_1(E_8)$, the Leech lattice VOA and the Moonshine VOA — are not classically free, disproving the widely held belief that every simple affine VOA at positive integer level is classically free. The conjecture remains open for general rational VOAs, including all holomorphic VOAs of CFT type.

Background

A vertex operator algebra (VOA) is an algebraic structure axiomatizing the chiral algebra of a two-dimensional conformal field theory: a complex vector space $V=\bigoplus_{n\ge0}V_n$ (an '$\mathbb{N}$-graded' VOA) with a state-field correspondence assigning to each $a\in V$ a formal field $Y(a,z)=\sum_{n\in\mathbb{Z}}a_nz^{-n-1}$ acting on $V$, subject to locality, vacuum, and Borcherds (Jacobi-type) identities. An admissible $V$-module is a $\mathbb{N}$-graded module $M=\bigoplus_{n\ge0}M_n$ on which the modes act compatibly with the grading, $a_nM_m\subseteq M_{m+\deg a-n-1}$; a VOA is called rational if every admissible module is a direct sum of simple admissible modules. Zhu associated to $V$ two quotient structures. Zhu's algebra $A(V)=V/O(V)$, where $O(V)$ is spanned by the bilinear products $a\circ b=\operatorname{Res}_z Y(a,z)b\,(1+z)^{\deg a}/z^2$, is an associative algebra whose irreducible modules biject with the irreducible admissible $V$-modules; Zhu proved that if $V$ is rational then $A(V)$ is finite-dimensional and semisimple. The $C_2$-algebra is $R_V=V/C_2(V)$, where $C_2(V)$ is the span of the mode products $u_{-2}v$; it is a commutative Poisson algebra with product $u_{-1}v$ and bracket $u_0v$, and $V$ is called $C_2$-cofinite when $\dim R_V<\infty$. The affine scheme $X_V=\operatorname{Specm}R_V$, the associated variety, is a fundamental invariant connecting the representation theory of $V$ with Poisson geometry. $C_2$-cofiniteness is the hypothesis of Miyamoto's modular invariance theorem and of the tensor-category and modularity framework for VOA module categories, and it drives finiteness of the module category; yet it is an internal condition on $V$, while rationality is an external semisimplicity condition, and relating the two is the crux of the subject.

The problem's lineage is explicit in the literature. Dong, Li and Mason introduced regularity — every weak module (a module with no grading assumed a priori) is a direct sum of irreducible ordinary modules — proved it for the standard rational families, and conjectured that rationality, $C_2$-cofiniteness and regularity are equivalent. Li proved that regular implies $C_2$-cofinite, and Abe, Buhl and Dong proved that for VOAs of CFT type ($\mathbb{N}$-graded with $V_0=\mathbb{C}\mathbf{1}$), rationality together with $C_2$-cofiniteness implies regularity; their paper calls the equivalence of rationality and $C_2$-cofiniteness 'one of the most important conjectures' in VOA theory. The direction $C_2$-cofinite $\Rightarrow$ rational is false — the triplet algebras of logarithmic conformal field theory are $C_2$-cofinite but not rational, as Gaberdiel and Gannon note — so the surviving open direction of the equivalence program is exactly Zhu's conjecture: rational $\Rightarrow$ $C_2$-cofinite; by the Abe–Buhl–Dong theorem this is equivalent, for CFT type, to asking whether every rational VOA is regular. A holomorphic VOA is a rational VOA that is its own unique simple admissible module; all non-trivial examples known (lattice VOAs of even self-dual lattices, the Moonshine VOA $V^{\natural}$ whose automorphism group is the Monster finite simple group) are $C_2$-cofinite by direct verification, but no general argument is known even in this most restrictive class.

The 2026 source paper of Song and Zeng adds a structural route. Li's filtration $F^pV$ (with $F^1V=C_2(V)$) makes the associated graded $\operatorname{gr}V$ a commutative differential algebra, and for any commutative algebra $R$ the jet algebra $J^{\infty}R$ — the differential algebra of infinite jets, i.e. of functions on the arc space of $\operatorname{Spec}R$ with derivation $\partial$ — is its universal differential-algebra cover. A VOA is classically free (a notion introduced by van Ekeren and Heluani for computations of chiral homology) when the canonical differential-algebra surjection $\pi:J^{\infty}R_V\twoheadrightarrow\operatorname{gr}V$ is an isomorphism. Separately, the canonical filtration of Zhu's algebra yields a natural surjection $\eta_V:R_V\twoheadrightarrow\operatorname{gr}A(V)$, known to be bijective when $V$ admits a PBW basis (an ordered monomial spanning set analogous to a Poincaré–Birkhoff–Witt basis; Arakawa–Lam–Yamada) but not in general. Song and Zeng prove that classical freeness forces $\eta_V$ to be an isomorphism; since $A(V)$ is finite-dimensional for rational $V$, this proves Zhu's conjecture on the classically free locus. The converse application is striking: for a non-trivial holomorphic VOA of CFT type, $A(V)=\mathbb{C}$ by Artin–Wedderburn, so classical freeness would force $R_V\cong\mathbb{C}$, hence $\operatorname{gr}V\cong J^{\infty}\mathbb{C}=\mathbb{C}$, and since Li's filtration is separated on an $\mathbb{N}$-graded VOA, $V=\mathbb{C}$ — a contradiction. Thus every non-trivial holomorphic VOA of CFT type is not classically free; examples include the lattice VOAs of the self-dual lattices $E_8$ and the Leech lattice, and the Moonshine VOA. This is consistent with Gaberdiel and Gannon's earlier computation that among root-lattice VOAs only $E_8$ is 'anomalous', i.e. has $\dim R_V>\dim A(V)$, and it disproves the conjecture (going back to Feigin's work on PBW filtrations) that every simple affine VOA $L_k(\mathfrak{g})$ — the irreducible quotient of the vacuum module for the affine Kac–Moody algebra at positive integer level $k$ — is classically free.

The empirical situation: every standard construction of rational VOAs (lattice VOAs, parafermion VOAs, diagonal cosets, orbifold fixed points, affine VOAs and $W$-algebras at admissible levels) has been verified to produce $C_2$-cofinite algebras, but always family by family — e.g. parafermion $C_2$-cofiniteness (Arakawa–Lam–Yamada), diagonal cosets (Lin), fixed-point subalgebras of regular VOAs (Carnahan–Miyamoto), and the icosahedral orbifold $V_{L_2}^{A_5}$ (Xu, August 2026). Recent structural work of Robert McRae proves results in the reverse direction (a simple self-contragredient $C_2$-cofinite VOA whose Zhu algebra is semisimple is rational) and states that even $C_2$-cofiniteness of cosets inside strongly rational VOAs is 'still not known in general'; the field uniformly defines 'strongly rational' as the conjunction of rationality and $C_2$-cofiniteness, precisely because no theorem deduces one from the other. Within the classically free world the positive results are now uniform (boundary Virasoro minimal models $\operatorname{Vir}_{2,2s+1}$, $L_k(\mathfrak{sp}_{2n})$ for all $k\ge1$, $L_1(\mathfrak{sl}_n)$), while the Ising model $\operatorname{Vir}_{3,4}$ and all non-trivial holomorphic CFT-type VOAs fail classical freeness — and it is exactly on that failure locus that Zhu's conjecture stands open.

Problem Statement

Zhu's conjecture (Conjecture 1.1 of arXiv:2606.14407, attributed to Zhu 1996): let $V$ be an $\mathbb{N}$-graded rational vertex operator algebra, i.e. $V=\bigoplus_{n\ge0}V_n$ and every admissible $V$-module is a direct sum of simple admissible $V$-modules. Is the $C_2$-algebra $R_V=V/C_2(V)$, where $C_2(V)=\operatorname{span}\{u_{-2}v:u,v\in V\}$, necessarily finite-dimensional — equivalently, is every rational VOA $C_2$-cofinite? By the Abe–Buhl–Dong theorem this is equivalent, for VOAs of CFT type, to asking whether every rational VOA is regular. The conjecture is proved for classically free VOAs (Song–Zeng 2026, via the isomorphism $R_V\cong\operatorname{gr}A(V)$) and holds by direct verification in every standard family of rational VOAs, but it is open in general — in particular for every non-classically-free rational VOA, which includes every non-trivial holomorphic VOA of CFT type. A complete answer is either a proof that $\dim R_V<\infty$ for all $\mathbb{N}$-graded rational $V$, or an explicit $\mathbb{N}$-graded rational VOA with $\dim R_V=\infty$. Adjacent questions — bijectivity of $\eta_V:R_V\twoheadrightarrow\operatorname{gr}A(V)$ beyond the classically free locus, and the classification of classically free vertex algebras — belong to the same program but are not required for closure of the conjecture itself.

References

1. Bailin Song and Xianlong Zeng, 'Zhu's algebra and the $C_2$-algebra of a classically free vertex operator algebra', arXiv:2606.14407 (2026), DOI 10.48550/arXiv.2606.14407, https://arxiv.org/abs/2606.14407
2. Yongchang Zhu, 'Modular invariance of characters of vertex operator algebras', Journal of the American Mathematical Society 9 (1996), 237–302, DOI 10.1090/S0894-0347-96-00182-8, https://doi.org/10.1090/S0894-0347-96-00182-8
3. Toshiyuki Abe, Geoffrey Buhl and Chongying Dong, 'Rationality, regularity, and $C_2$-cofiniteness', Transactions of the American Mathematical Society 356 (2004), 3391–3402, DOI 10.1090/S0002-9947-03-03413-5, preprint arXiv:math/0204021, https://arxiv.org/abs/math/0204021
4. Chongying Dong, Haisheng Li and Geoffrey Mason, 'Regularity of rational vertex operator algebras', Advances in Mathematics 132 (1997), 148–166, DOI 10.1006/aima.1997.1681, https://doi.org/10.1006/aima.1997.1681
5. Haisheng Li, 'Some finiteness properties of regular vertex operator algebras', Journal of Algebra 212 (1999), 495–514, DOI 10.1006/jabr.1998.7654, https://doi.org/10.1006/jabr.1998.7654
6. Matthias R. Gaberdiel and Terry Gannon, 'Zhu's algebra, the $C_2$ algebra, and twisted modules', Contemporary Mathematics 497 (2009), 65–78, DOI 10.1090/conm/497/09769, preprint arXiv:0811.3892, https://arxiv.org/abs/0811.3892
7. Jethro van Ekeren and Reimundo Heluani, 'The first chiral homology group', Communications in Mathematical Physics 405 (2024), Paper No. 194, DOI 10.1007/s00220-024-05061-z, arXiv:2103.06322, https://arxiv.org/abs/2103.06322
8. Andrew R. Linshaw and Bailin Song, 'Cosets of free field algebras via arc spaces', International Mathematics Research Notices IMRN 2024, no. 1 (2024), 47–114, DOI 10.1093/imrn/rnac367, arXiv:2109.09050, https://arxiv.org/abs/2109.09050
9. Robert McRae, 'On rationality for $C_2$-cofinite vertex operator algebras', Cambridge Journal of Mathematics 14 (2026), no. 1, 1–115, DOI 10.4310/cjm.260225023743, preprint arXiv:2108.01898, https://arxiv.org/abs/2108.01898
10. Xingjun Lin, 'Rationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras', Journal of Algebra 608 (2022), 645–672, DOI 10.1016/j.jalgebra.2022.07.001, arXiv:2107.09400, https://arxiv.org/abs/2107.09400
11. Evgeny Feigin, 'The PBW filtration', Representation Theory 13 (2009), 165–181, DOI 10.1090/s1088-4165-09-00349-5, arXiv:math/0702797, https://arxiv.org/abs/math/0702797
12. Tomoyuki Arakawa, Ching Hung Lam and Hiromichi Yamada, 'Zhu's algebra, $C_2$-algebra and $C_2$-cofiniteness of parafermion vertex operator algebras', Advances in Mathematics 264 (2014), 261–295, arXiv:1207.3909, https://arxiv.org/abs/1207.3909
13. Scott Carnahan and Masahiko Miyamoto, 'Regularity of fixed-point vertex operator subalgebras', arXiv:1603.05645, https://arxiv.org/abs/1603.05645
14. Masahiko Miyamoto, 'Modular invariance of vertex operator algebras satisfying $C_2$-cofiniteness', Duke Mathematical Journal 122 (2004), 51–91, DOI 10.1215/s0012-7094-04-12212-2, https://doi.org/10.1215/s0012-7094-04-12212-2
15. Shun Xu, '$C_2$-Cofiniteness and Rationality of the Icosahedral Orbifold $V_{L_2}^{A_5}$', arXiv:2608.13872 (2026), https://arxiv.org/abs/2608.13872