Quasi-Lisse Simple Affine Vertex Algebras

• Quasi-lisse simple affine vertex algebras are defined by finiteness conditions on their associated Poisson varieties, which yield rigid module categories and MLDEs for character theory.
• These algebras are characterized by the geometric classification via closures of nilpotent orbits and arise at both admissible and special non-admissible levels.
• Their study connects representation theory, algebraic geometry, and physics through explicit constructions in affine, orbifold, and coset models with modular properties.

Quasi-lisse simple affine vertex algebras are an important class in the modern theory of vertex operator algebras (VOAs), distinguished by geometric, representation-theoretic, and modular properties that generalize those of admissible affine vertex algebras. Defined by a finiteness condition on their associated (Poisson) varieties, these algebras exhibit rigid behavior in their module categories and connections to modular linear differential equations, as well as deep links to symplectic and geometric representation theory.

1. Definitions: Vertex Algebras, Associated Variety, and the Quasi-Lisse Condition

A vertex algebra $V$ is a complex vector space equipped with a vacuum vector $|0\rangle$ and a state–field correspondence $Y(-,z): V \to \mathrm{End}\,V[[z,z^{-1}]]$, satisfying locality, translation covariance, and vacuum axioms. A grading by a Hamiltonian $H$ decomposes $V$ as $$V = \bigoplus_{\Delta \in \frac{1}{r_0} \mathbb{N}_0} V_\Delta$$ with $V_0 = \mathbb{C}|0\rangle$; $V$ is said to be conical if this grading is positive and $V_0$ is one-dimensional.

For a vertex algebra $V$, the $C_2$-algebra is $R_V := V / F^1 V$ where $$F^1 V := \mathrm{span}\{a_{(-2)}b \mid a, b \in V\}$$, and $R_V$ inherits a Poisson algebra structure. The associated variety is $X_V := \mathrm{Specm}\, R_V$, a conical affine Poisson variety.

A conformal vertex algebra $V$ is quasi-lisse if $X_V$ has finitely many symplectic leaves; equivalently, $R_V$ has finitely many symplectic cores. This property strictly generalizes $C_2$-cofiniteness (the lisse property, which requires $\dim X_V = 0$). For simple affine VOAs $L_k(\mathfrak{g})$, $X_{L_k(\mathfrak{g})}$ equals the closure of a nilpotent orbit for suitable $k$. Quasi-lisse VOAs admit only finitely many simple ordinary modules and their module characters satisfy modular linear differential equations (MLDEs) (Arakawa et al., 2016).

2. Structure of Quasi-Lisse Simple Affine Vertex Algebras

Let $\mathfrak{g}$ be a simple Lie algebra with dual Coxeter number $h^\vee$. The universal affine VOA $V^k(\mathfrak{g})$ at level $k \in \mathbb{C} \setminus \{-h^\vee\}$ has a simple quotient $L_k(\mathfrak{g}) = V^k(\mathfrak{g})/I_k$; for $k \neq -h^\vee$, it is conical, simple, and self-dual.

A sharp structural dichotomy emerges for $L_k(\mathfrak{g})$:

• If $k$ is an admissible rational number (per Kac–Wakimoto), $L_k(\mathfrak{g})$ is quasi-lisse and $X_{L_k(\mathfrak{g})}$ is the Zariski closure of a single nilpotent orbit in $\mathfrak{g}^*$, explicitly determined by the parameters of admissibility (Arakawa et al., 2016).
• There exist non-admissible $k$ (e.g., $L_{-2}(G_2)$, $L_{-2}(B_3)$) where $L_k(\mathfrak{g})$ is quasi-lisse with associated variety equal to the closure of the minimal nilpotent orbit (Arakawa et al., 2015, Adamović et al., 18 Apr 2025).

The associated geometry is rigid: ordinary modules cannot realize proper subvarieties inside $X_V$ if $X_V$ is irreducible.

3. Ordinary Modules and Rigidity of Associated Varieties

An ordinary module $M$ for a vertex algebra $V$ admits an $H$-grading by generalized eigenvalues with finite-dimensional graded pieces; the translation operator (derivation) acts semisimply. For each such module, the associated variety $X_M \subset X_V$ is defined analogously via $\bar M = M / F^1 M$.

For a conical, simple, self-dual, quasi-lisse vertex algebra $V$ and any simple ordinary module $M$:

• $\dim X_M = \dim X_V$,
• If $X_V$ is irreducible, then $X_M = X_V$ (Villarreal, 4 Nov 2025).

This is established via a sequence of fusion-ideal bounds: the existence of a surjective intertwining operator $I: M_1 \otimes M_2 \to M_3\{z\}$ implies $$\dim X_{M_3} \leq \min(\dim X_{M_1}, \dim X_{M_2}) + 1$$. Applying this to the self-duality structure enforces that $\dim X_M = \dim X_V$, and by parity constraints (quasi-lisse associated varieties have even dimension), no smaller symplectic leaves can be realized in the geometry of ordinary modules.

This rigidity drastically simplifies the representation theory: all simple ordinary modules for $L_k(\mathfrak{g})$ (quasi-lisse) have associated variety equal in dimension to that of $L_k(\mathfrak{g})$, and if $X_{L_k(\mathfrak{g})}$ is irreducible, the associated varieties coincide exactly.

4. Classification and Geometry of Quasi-Lisse Simple Affine Examples

The prototypical quasi-lisse simple affine vertex algebras arise at admissible levels and at special (Deligne, boundary, or exceptional) levels:

• Admissible levels: For $k = p/q$ admissible, $X_{L_k(\mathfrak{g})} = \overline{\mathcal{O}_k}$, the closure of a single nilpotent orbit, irreducible of even (known) dimension (Arakawa et al., 2016, Villarreal, 4 Nov 2025).
• Non-admissible cases (Deligne exceptional chain): For $k = -h^\vee/6-1$ and $\mathfrak{g}$ in the Deligne exceptional series, $X_{L_k(\mathfrak{g})}$ is the closure of the minimal nilpotent orbit. For $D_4$, levels $k_m = -6 + 4/(2m+1)$ (e.g., $k_0 = -2$, $k_1 = -14/3$ (Adamović et al., 18 Apr 2025)), the associated variety is contained in the nilpotent cone, and explicit computational verification establishes quasi-lisse property for $m=0,1$.
• Quasi-lisse extensions: Certain orbifolds, cosets, and infinite simple-current extensions, such as $\mathrm{FT}_p(\mathfrak{sl}_2)$, $U(n)=\mathrm{psl}(n|n)_1/\mathfrak{sl}(n)_1$, and $C_p$-series, are shown or conjectured quasi-lisse via analysis of their characters, modular properties, and associated varieties (Creutzig et al., 2023, Adamovic et al., 2018, Adamovic et al., 3 Feb 2025).

A table summarizing some of these examples:

Algebra	Level	Associated Variety
$L_k(\mathfrak{g})$ (admissible)	$k = p/q$ admissible	Closure of nilpotent orbit $\mathcal{O}_k$
$L_{-2}(G_2)$, $L_{-2}(B_3)$	Non-admissible negative integer	Minimal nilpotent orbit closure
$L_{-14/3}(D_4)$	$-6 + 4/3$	Nilcone (computational confirmation)
$\mathrm{FT}_p(\mathfrak{sl}_2)$	$k = -2+1/p$	Nilpotent cone of $\mathfrak{sl}_2$
$U(n)$ (coset)	$k = -1$	Projected: finite symplectic leaf variety
$C_p$ (family with conformal embeddings)	$k_1=-2+1/p$, $k_2=-2-1/p$	Supported by modular/character analysis

5. Modular Linear Differential Equations and Character Theory

A defining feature of quasi-lisse VOAs is the modularity property of characters: the normalized character of any simple ordinary module satisfies a modular linear differential equation (MLDE). Arakawa–Kawasetsu established that for any quasi-lisse VOA $V$ of central charge $c$, each normalized character $\chi_M(\tau)$ extends holomorphically to $\mathbb{H}$ and satisfies an MLDE:

$$\left(D^N + \sum_{i=0}^{N-1} \phi_i(\tau) D^i\right) \chi_M(\tau) = 0,$$

where $D$ is the Serre/Ramanujan-Serre derivative and $\phi_i$ are (quasi-)modular forms (Arakawa et al., 2016).

For the Deligne exceptional series, the vacuum character of $L_k(\mathfrak{g})$ satisfies a second-order MLDE with explicit modular coefficient functions, and explicit closed forms for these characters are given in terms of eta-quotients, modular forms, and Eisenstein series. These match the homogeneous Schur indices of corresponding 4d SCFTs.

In more generality, the modular theory extends to twisted modules, spectral flows, and orbifolds, with the modular group potentially replaced by congruence subgroups depending on the twist (e.g., $\Gamma^0(2)$ for $\mathbb{Z}_2$-twisted modules) (Li et al., 2023).

6. Representation-Theoretic and Geometric Implications

The geometry of the associated variety tightly controls representation theory:

• For quasi-lisse $V$, all simple ordinary modules have maximal associated variety dimension, and—if $X_V$ is irreducible—every such module "sees" the full symplectic leaf structure.
• Quasi-lisse property ensures only finitely many simple ordinary modules exist.
• For minimal W-algebras arising from quantum Drinfeld–Sokolov reduction at quasi-lisse levels, lisse (i.e., $C_2$-cofinite) structure occurs if the corresponding affine vertex algebra's associated variety realizes the minimal nilpotent orbit closure (Arakawa et al., 2015).
• In explicit families such as $L_{k_m}(D_4)$, explosion in the number of category–$\mathcal{O}$ irreducibles is observed at higher $m$, but remains a unique irreducible ordinary module at each level proven to be quasi-lisse (Adamović et al., 18 Apr 2025).

The quasi-lisse condition enables classification of affine, orbifold, and coset VOAs in terms of associated symplectic geometry and modular data, suggesting completeness for known examples at admissible and certain exceptional levels.

7. Open Problems and Outlook

Current open problems in the area include:

• Proving quasi-lisse property for all conjectured infinite series, such as $L_{k_m}(D_4)$ at arbitrary $m > 1$, by structural rather than computational methods.
• Determining precise associated varieties in more general families, particularly in higher-rank analogues of $C_p$ and related orbifolds.
• Understanding the full extent of modularity properties for generalized (logarithmic, twisted) modules in the quasi-lisse context.

Recent progress continues to reinforce the unifying perspective that quasi-lisse simple affine vertex algebras are characterized by the finiteness of their symplectic leaf stratification and the rigidity of their module categories, with broad significance for algebraic geometry, number theory, and physics-driven representation theory.