Converses to modular invariance of characters: do modular-invariant characters force admissibility, rationality, and simplicity?

Kac and Wakimoto's paper 'On modular invariance of quantum affine W-algebras' (Commun. Math. Phys. 406, 2025) proves that certain quantum affine W-algebras at non-admissible levels — minimal W-algebras of types $D_n$, $E_6$, $E_7$, $E_8$ at negative integer levels, and Hamiltonian reductions $H_f(k\Lambda_0)$ at principal admissible levels — are modular invariant, but its structural conclusions are conditional on three unproved converse conjectures: (1) the 1989 Kac–Wakimoto conjecture that normalized characters of irreducible highest-weight modules over an affine Lie algebra are modular invariant only for admissible modules; (2) the corrected Dong–Ren conjecture that a modular-invariant lisse ($C_2$-cofinite) conformal vertex algebra is rational; (3) the conjecture that a modular-invariant conformal vertex algebra is simple. This problem asks to prove or refute these three converses, which jointly ask whether modular behaviour of characters characterizes the algebraically regular classes.

Background

Let $\mathfrak g$ be a complex simple finite-dimensional Lie algebra with highest root $\theta$, and let $\hat{\mathfrak g}=\mathfrak g[t,t^{-1}]\oplus\mathbb C K\oplus\mathbb C d$ be the associated affine Lie algebra (a central extension of the polynomial loop algebra); the level of a highest-weight $\hat{\mathfrak g}$-module $L(\Lambda)$ is the scalar $k=\Lambda(K)$, while $h^\vee$ denotes the dual Coxeter number (the standard Lie-theoretic invariant normalizing the invariant bilinear form) and $r^\vee$ the lacing number (the maximal number of edges joining two nodes of the Dynkin diagram of $\mathfrak g$). The normalized character of $L(\Lambda)$ is $\mathrm{tr}_{L(\Lambda)}q^{L_0-c/24}$, viewed as a function of $q=e^{2\pi i\tau}$ on the complex upper half-plane (with extra variables $z,t$ in the full affine setting), where $c$ is the Virasoro central charge. Kac and Peterson proved that for integrable (dominant integral) levels these characters are holomorphic modular functions whose span is $SL_2(\mathbb Z)$-invariant. Kac and Wakimoto then introduced the admissible highest weights — the class containing the integrable ones for which the Weyl–Kac character formula still holds; a level $k$ is principal admissible if $k+h^\vee=p/u$ with $\gcd(p,u)=\gcd(u,r^\vee)=1$ and $p\ge h^\vee$ — extended the modular transformation laws to admissible modules, and conjectured at the same time (1989) that admissible modules are the only irreducible highest-weight modules with modular behaviour.

On the vertex-algebra side, a vertex operator algebra (VOA, the algebraic object encoding the chiral symmetry of a two-dimensional conformal field theory) carries a graded character $\mathrm{tr}_V q^{L_0-c/24}$; following Kac–Wakimoto, a conformal VOA is called modular invariant if this character converges to a holomorphic modular function on a congruence subgroup of $SL_2(\mathbb Z)$ (a subgroup of finite index containing the principal congruence subgroup $\Gamma(N)$, the matrices congruent to the identity modulo $N$, for some $N$). Zhu's theorem gives the forward direction: for a rational VOA (finitely many irreducible modules, every module completely reducible) that is lisse, i.e. $C_2$-cofinite ($\dim_\mathbb C V/C_2(V)<\infty$, where $C_2(V)$ is the span of all products $a_{(-2)}b$), the characters of the irreducible modules span a finite-dimensional representation of the modular group.

Quantum Hamiltonian reduction attaches to a non-zero nilpotent element $f\in\mathfrak g$ a BRST-type functor $H_f$ on affine VOAs and their modules, producing the affine W-algebras $W^k(\mathfrak g,f)$ (universal) and $W_k(\mathfrak g,f)$ (simple quotient); for the minimal nilpotent element $f_{min}=e_{-\theta}$ these are the minimal W-algebras $W^{min}_k(\mathfrak g)$, and one writes $\widetilde W_k(\mathfrak g,f):=H_f(k\Lambda_0)$. Kawasetsu discovered that $W^{min}_{-h^\vee/6}(\mathfrak g)$ is rational for $\mathfrak g=D_4,E_6,E_7,E_8$ (the types of the Deligne exceptional series, a distinguished list of Lie algebras with uniform representation rings) even though the level $-h^\vee/6$ is not admissible; Arakawa and Moreau proved these algebras are lisse for all negative integer $k\ge -h^\vee/6$ (and for $D_n$, $n>4$, also for $k=-1,-2$); Arakawa–Creutzig–Kawasetsu conjectured rationality of the whole family. The paper [1] proves that these $W^{min}_k(\mathfrak g)$, and the reductions $\widetilde W_k(\mathfrak g,f)$ for any non-zero nilpotent $f$ with Dynkin characteristic $2x$ (the labels of the neutral element $x$ of the associated $sl_2$-triple on the simple roots, which determine the nilpotent orbit) at a principal admissible level of denominator $u>\theta(x)$, are modular invariant. However, the structural payoff the authors want — rationality of these W-algebras and the identification $\widetilde W_k(\mathfrak g,f)=W_k(\mathfrak g,f)$ — is only conditional: the paper's abstract itself states that modular invariance 'conjecturally … implies that V is simple, and that V is rational, provided that it is lisse'. All known theorems run from structure to modularity; the converses, which are the content of this problem, are open.

Problem Statement

The problem is to settle the converses of the classical modularity theorems for affine Lie algebra characters and conformal vertex algebras — three linked conjectures, each of which must be proved in the stated generality or refuted by an explicit, verified counterexample.

1. (Kac-Wakimoto 1989, 'only admissible' conjecture.) Let $\hat{\mathfrak g}$ be an affine Lie algebra over $\mathbb C$, with $\overline{\mathfrak g}$ the underlying finite-dimensional simple Lie algebra in the non-twisted case. Every irreducible highest-weight $\hat{\mathfrak g}$-module $L(\Lambda)$ at a non-critical level ($k\neq -h^\vee$, where normalized characters are defined) whose normalized character — normalized by a power of $q=e^{-\delta}$, with $\delta$ the null root, and viewed on the domain $\{h\in\mathfrak h\mid \mathrm{Re}\,\delta(h)>0\}$ with coordinates $(\tau,z,t)$ — is a meromorphic modular function, is admissible: the admissible modules are all the irreducible highest-weight modules with this property (for admissible modules the modular transformation laws close over the finite family $\hat{Pr}^k \bmod \mathbb C\delta$). In the non-twisted vertex-algebra specialization, Kac-Wakimoto state the corresponding conjecture for the vacuum character (Conjecture 0.1 of arXiv:2504.17159): $\mathrm{tr}_{V_k(\overline{\mathfrak g})}e^{2\pi i(L_0-C_k/24)\tau}$, with $C_k=\frac{k\dim\overline{\mathfrak g}}{k+h^\vee}$, converges to a holomorphic modular function on the upper half-plane if and only if $k$ is a principal or subprincipal admissible level (the 'if' direction is known). This vacuum-character conjecture is not equivalent to the module-level statement above — it constrains only the vacuum module $L(k\Lambda_0)$ and restricts to holomorphic, rather than meromorphic, modularity — so settling it is progress toward, but does not by itself complete, Part 1. Prove the module-level conjecture, or exhibit a non-admissible irreducible highest-weight module whose normalized character has the stated modular property.

2. (Corrected Dong-Ren conjecture.) Every modular-invariant conformal vertex algebra that is lisse ($C_2$-cofinite) is rational. Prove this, or exhibit a modular-invariant lisse vertex algebra that is not rational (for example, one with infinitely many irreducible modules or a non-semisimple module category).

3. (Simplicity.) Every modular-invariant conformal vertex algebra is simple, i.e. coincides with its simple quotient. Prove this, or exhibit a non-simple modular-invariant conformal vertex algebra.

The common scientific objective is to determine whether modular behaviour of characters is by itself a complete regularity criterion — whether the analytic property of modularity forces the algebraic regularity (admissibility in the affine case; rationality under lisseness, and simplicity, in the vertex algebra case). The three parts are linked exactly as in the paper [1]: conjecture 2 converts the proven modularity of the lisse W-algebras $W^{min}_k(\mathfrak g)$ at negative integer levels into their rationality (the Arakawa-Creutzig-Kawasetsu conjecture), conjecture 3 identifies the quantum reductions $\widetilde W_k(\mathfrak g,f)=H_f(k\Lambda_0)$ with the simple W-algebras $W_k(\mathfrak g,f)$, and conjecture 1 is the affine-representation-theoretic ancestor of the same philosophy that the whole program rests on. The problem is complete when each of the three conjectures has been settled (each either proved in the stated generality or refuted by an explicit counterexample); results restricted to particular Lie types, levels, or subclasses of vertex algebras are progress but do not complete it.

References

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