Conical Simple Self-Dual Quasi-Lisse Vertex Algebra

• Conical simple self-dual quasi-lisse vertex algebra is a vertex (super)algebra characterized by conical grading, simplicity, self-duality, and a quasi-lisse associated variety.
• The algebra imposes strict constraints on representation theory through its C₂-filtration, fusion rules, and Poisson geometric structure.
• Key applications include affine and N=4 supersymmetric vertex algebras, linking algebraic theory with 4d/2d dualities and symplectic singularity studies.

A conical simple self-dual quasi-lisse vertex algebra is a vertex (super)algebra $V$ distinguished by the intersection of four key properties: conical grading, simplicity, self-duality, and quasi-lisse associated variety. Such vertex algebras are central objects in the paper of algebraic, geometric, and physical aspects of representation theory, symplectic singularities, and 4d/2d dualities in supersymmetric quantum field theory. These properties impose strong constraints on their representation theory and ensure remarkable geometric uniformity across their ordinary modules, as formalized in recent results on associated varieties.

1. Definitions and Structural Hypotheses

Let $(V,Y(\,{\cdot}\,,z),\mathbf{1})$ be a vertex algebra equipped with a Hamiltonian $H$, and with a grading by conformal (or Hamiltonian) weight: $$V = \bigoplus_{\Delta \in \frac{1}{r_0}\mathbb{Z}_{\geq0}} V_\Delta$$ for some positive integer $r_0$. The defining properties are as follows:

• Conical: $V_0 = \mathbb{C}$ and $V$ is positively graded; the associated variety $X_V$ admits a contracting $\mathbb{C}^\times$-action rescaling the Poisson bracket, and $\mathcal{O}(X_V)$ has only nonnegative $\mathbb{C}^\times$-weights with $\mathbb{C}$ as the weight-zero piece.
• Simple: $V$ contains no proper nontrivial vertex ideals; equivalently, $V$ is irreducible as a module over itself.
• Self-dual: The restricted dual $V' = \bigoplus_\Delta V_\Delta^*$ is isomorphic to $V$ as a module, i.e., $V \cong V'$.
• Quasi-lisse: The associated variety $X_V$ has only finitely many symplectic leaves (i.e., finitely many Poisson leaf strata), which is equivalent to the condition that $V$ satisfies certain finiteness properties analogous to those of lisse (smooth) algebras, but allowing for symplectic singularities.

This suite of conditions severely restricts the algebraic and geometric structure of $V$ and its category of ordinary modules.

2. Associated Varieties and the $C_2$ Filtration

The associated variety provides a geometric invariant for both vertex algebras and their modules. It is constructed as follows:

• The $C_2$-filtration on $V$ is defined by $$F^1 V = \operatorname{Span}\{u_{(-2)}v \mid u,v \in V\}$$, and for a module $M$, $$F^1 M = \operatorname{Span}\{u_{(-2)}m\mid u\in V, m\in M\}$$.
• One sets $R_V = V/F^1V$, which acquires a Poisson algebra structure, and $\overline M = M/F^1M$, a Poisson module over $R_V$.
• The associated variety of $V$ is $X_V = \operatorname{Specm}(R_V)$; for a $V$-module $M$, the associated variety is $$X_M = \operatorname{Supp}_{R_V}(\overline M) \subset X_V$$.

For modules, the associated variety may alternatively be characterized using an increasing filtration $\Gamma^pM$ (with $\Gamma^0M = M$ and $\Gamma^1M = F^1M$), the associated graded $\operatorname{gr} M$, and $$X_M = \operatorname{Supp}_{R_V}(\operatorname{gr} M)$$.

An ordinary $V$-module is a positive-energy module: $$M = \bigoplus_{\Delta \in h + \frac{1}{r_0}\mathbb{Z}_{\geq 0}} M_\Delta,\quad \dim M_\Delta < \infty$$ with the property that $\overline M$ is finitely generated over $R_V$.

3. Main Theorem on Uniformity of Associated Varieties

The central result for conical simple self-dual quasi-lisse vertex algebras states:

Let $V$ be such a vertex algebra, and $M$ any simple ordinary $V$-module. Then: $> \dim X_M = \dim X_V >$ Moreover, if $X_V$ is irreducible as a variety, then $> X_M = X_V >$ for every simple ordinary $M$.

The proof incorporates a combinatorial analysis of module fusion and duality, as well as the even-dimensionality of symplectic leaves in the quasi-lisse case. Key components are as follows:

• Fusion Dimension Bound (Prop 3.3): For ordinary modules $M_1$, $M_2$, $M_3$ with a surjective intertwiner of type $\binom{M_3}{M_1\,M_2}$,

$$\dim X_{M_3} \leq \min\{\dim X_{M_1}, \dim X_{M_2}\} + 1$$

by analyzing Borcherds identities and highest-weight inclusions.

• Self-Dual Bound (Prop 3.4): Applying duality to module intertwiners yields

$$\dim X_V \leq \dim X_M + 1,\qquad \dim X_M \leq \dim X_V + 1$$

which, together with $X_M \subset X_V$, forces $\dim X_M \in \{\dim X_V,\, \dim X_V-1\}$.

• Even Codimension Step: The quasi-lisse property entails all irreducible components of $X_V$ have even dimension. As $X_M$ is a closed Poisson-stable subvariety of $X_V$, its dimension cannot differ from $\dim X_V$ by one, so equality is forced.

If $X_V$ is further irreducible, $X_M = X_V$ is the only possibility.

4. Examples and Applications: Affine and Supersymmetric Vertex Algebras

The theorem applies to numerous important families:

• Simple affine vertex algebras $L_k(\mathfrak{g})$:
  • For admissible levels $k$, $L_k(\mathfrak{g})$ is quasi-lisse and $X_{L_k}$ is the closure of a single nilpotent orbit. Then $X_M = X_{L_k}$ for every ordinary simple module $M$.
  • Recent results extend quasi-lisse and irreducibility properties to non-admissible $k$ (e.g., $k=-2$ for $\mathfrak{g}=G_2, B_3$), implying the same conclusion for $X_M$.
• $\mathcal{N}=4$ supersymmetric vertex operator algebras $W_{S_N}$:
  • $W_{S_N}$, constructed via BRST reduction and free-field realizations over Hilbert schemes of points, is a (conjecturally) simple, conical, self-dual, quasi-lisse vertex algebra for $N\geq 2$.
  • The associated variety $X_{W_{S_N}}$ is the symplectic singularity $\mathcal{M}_{S_N}$, and the $\mathbb{C}^\times$-action provides conicality. For $N=2$, this is the simple small $\mathcal{N}=4$ algebra at $c=-9$.
  • The same methodologies generalize to certain Nakajima quiver varieties and other nonclassical settings (Arakawa et al., 2023).

5. Implications in Representation Theory and Physics

The uniformity of associated varieties across all ordinary modules over conical simple self-dual quasi-lisse $V$ has deep consequences:

• Representation theory: Geometric invariants (such as characteristic varieties and D-module supports) coincide within the block of ordinary modules, providing constraints on the possible behavior of modules and their physical or geometric realizations.
• Geometric correspondence: The associated variety $X_V$ frequently coincides with the Higgs branch of 4d $\mathcal{N}=2$ superconformal field theories whose 2d chiral algebras are $V$. The theorem implies that (ordinary) line- or surface-operator sectors "see" the same Higgs branch geometry.
• Fusion rules and tensor categories: The geometric uniformity supports a systematic approach to fusion and tensor-category theory within the category of ordinary modules, with connections to symplectic singularity theory, modular linear differential equations, and noncommutative resolutions.

6. Character, Modular Properties, and Further Directions

For explicit classes (e.g., $W_{S_N}$):

• Characters: Supercharacters may be computed via Euler–Poincaré analysis and matrix integrals. For $W_{S_N}$,

$$\operatorname{sch}_{W_{S_N}}(q) = \frac{1}{\eta(q)^3}\sum_{n=0}^\infty (-1)^n\left[\binom{N+n}{N} + \binom{N+n-1}{N}\right]q^{(N+2n)^2/8}$$

which match (after normalization) the Schur indices of associated 4d $\mathcal{N}=4$ super Yang–Mills theories.

• Modularity: These characters are holomorphic quasimodular forms, with weights and congruence subgroups determined by $N$ parity; e.g., for $N$ odd, $SL_2(\mathbb{Z})$, for even $N$, $\Gamma^0(2)$.
• Generalizations: The techniques extend to vertex algebras attached to other symplectic singularities, including Nakajima quiver varieties, tying the theory to diverse fields such as geometric representation theory, symplectic geometry, and mathematical physics.

7. Summary Table of Defining Features

Property	Definition (in this context)	Example(s)
Conical	$V_0 = \mathbb{C}$, positive grading, contracting $\mathbb{C}^\times$-action	$L_k(\mathfrak{g})$, $W_{S_N}$
Simple	No nonzero proper vertex-ideals	$L_k(\mathfrak{g})$ for admissible $k$
Self-dual	$V \cong V'$ as $V$-modules (restricted duality)	$W_{S_N}$, small $\mathcal{N}=4$ ($N=2$)
Quasi-lisse	$X_V$ has finitely many symplectic leaves	$L_k(\mathfrak{g})$, $W_{S_N}$

The confluence of these properties in a vertex algebra tightly constrains the geometry of associated varieties across ordinary modules, facilitating the paper of their representation categories and links to physics, symplectic geometry, and algebraic geometry (Villarreal, 4 Nov 2025, Arakawa et al., 2023).