Vertex Algebra of Chiral Differential Operators

• Vertex Algebra of Chiral Differential Operators is a dg extension combining vertex algebra techniques with derived geometry to refine invariants on singular moduli spaces.
• The construction employs BRST reduction and mirror involutions to embed functions into a chiral de Rham complex that models derived critical loci.
• This approach unifies methods from derived algebraic geometry, BV quantization, and equivariant localization to compute key invariants and graded characters.

The vertex algebra of chiral differential operators (CDOs) on derived critical loci is a differential graded (dg) extension of the theory of CDOs, providing a chiral, locally conformal field theoretical, and cohomological refinement of geometric invariants on singular moduli spaces. Central to their construction is the application of vertex operator algebra (VOA) techniques, BRST reduction, derived algebraic geometry, and Batalin–Vilkovisky (BV) quantization to the study of the derived critical locus $\mathbf{crit}_f$ of a function $f$ on a smooth algebraic variety $X$. The resulting dg-VOA $\mathbf{crit}^{ch}_f$ realizes the chiral symmetries and topological features of $N=2$ Landau–Ginzburg models and serves as a bridge linking derived geometry, vertex algebras, and mathematical physics (Bouaziz, 2024).

1. Construction of the dg-VOA on Derived Critical Loci

Given a smooth $D$-dimensional complex algebraic variety $(X, \operatorname{vol})$ with nowhere vanishing volume form and regular function $f \in \mathcal{O}_X$, the derived critical locus $\mathbf{crit}_f = \{df = 0\}_{\mathrm{der}}$ carries a dg structure sheaf

$$\mathcal{O}(\mathbf{crit}_f) \simeq (\wedge^\bullet T_X, d_f)$$

where $f$0 is the tangent sheaf and the differential $f$1 is the Schouten bracket contraction with $f$2.

The chiral de Rham complex $f$3 of Malikov–Schechtman–Vaintrob is a sheaf of topological VOAs whose weight-zero subspace is $f$4. Twisting by the mirror involution exchanges forms and polyvectors, yielding a sheaf $f$5 of VOAs with weight-zero subspace $f$6. A function $f$7 embeds as a weight-zero field. Using the odd current $f$8 of weight 2, the field $f$9 (weight 1, polyvector-degree $X$0) enables BRST reduction: $X$1 At weight zero, $X$2 reproduces the Koszul (cohomological) model for the classical derived critical locus.

2. G$X$3-Equivariant Homogeneity and Topological Structure

If $X$4 admits a $X$5-action with $X$6 of weight $X$7 and $X$8 of weight $X$9, there is an induced grading. Under these hypotheses, $\mathbf{crit}^{ch}_f$0 acquires a topological VOA structure:

• There exist four BRST-closed currents $\mathbf{crit}^{ch}_f$1 corresponding to Virasoro, Heisenberg, and odd $\mathbf{crit}^{ch}_f$2 superconformal generators, satisfying the OPEs of the topological VOA $\mathbf{crit}^{ch}_f$3 with

$\mathbf{crit}^{ch}_f$4

This "topological rank" is determined by simple representation-theoretic manipulations involving the shift of $\mathbf{crit}^{ch}_f$5 by $\mathbf{crit}^{ch}_f$6, where $\mathbf{crit}^{ch}_f$7 is the infinitesimal generator of the $\mathbf{crit}^{ch}_f$8-action. Central charge computations confirm that the Heisenberg algebra generated by $\mathbf{crit}^{ch}_f$9 has the expected rank (Bouaziz, 2024).

3. Vertex Algebra, BRST Reduction, and OPE Structure

The VOA structure on $N=2$0 is that of a dg vertex algebra arising via BRST reduction:

• The cohomology at conformal weight zero recovers the exterior algebra of polyvector fields on $N=2$1 with differential $N=2$2.
• The full VOA includes fields of varying conformal weights and polyvector degrees, with operator products reflecting the algebraic structure of both the tangent algebra and the derived geometry.
• The key structural fields satisfy OPEs corresponding to Virasoro and Heisenberg algebras and the relations expected in topological $N=2$3 systems at central charge $N=2$4.

4. Batalin–Vilkovisky Quantization and Cohomological Properties

The derived critical locus has a canonical $N=2$5-shifted symplectic (dg-Poisson) structure. BV quantization proceeds as follows:

• The algebra $N=2$6 admits a BV quantization: a second-order differential operator $N=2$7 (with divergence operator $N=2$8 fixed by volume trivialization) and quantized complex

$N=2$9

• The chiral lift is

$D$0

The natural inclusion of the non-chiral BV complex into the chiral complex is a quasi-isomorphism, collapsing via a spectral sequence whose differentials are controlled by conformal weight and $D$1 structure (Bouaziz, 2024).

5. Graded Characters, Localization, and Invariants

Characters of chiral critical locus VOAs encode enumerative invariants:

• The doubly-graded trace (character) is

$D$2

where $D$3 and $D$4 track conformal weight and polyvector degree, respectively.

• For typical examples (e.g., $D$5 the total space of a canonical bundle $D$6 over a projective $D$7-fold $D$8 with isolated fixed points under $D$9), equivariant localization computes

$(X, \operatorname{vol})$0

where $(X, \operatorname{vol})$1 are weights of tangent action and $(X, \operatorname{vol})$2 is the theta function. This formula encodes both classical and quantum cohomological content (e.g., deformed Euler characteristics and indices).

6. Connections to N=2 Landau–Ginzburg Models and Applications

When $(X, \operatorname{vol})$3, $(X, \operatorname{vol})$4 reduces to the (possibly twisted) chiral de Rham complex; for general $(X, \operatorname{vol})$5, it provides a chiral realization of the category of matrix factorizations in Landau–Ginzburg models. In physics, the VOA $(X, \operatorname{vol})$6 coincides with the chiral algebra of $(X, \operatorname{vol})$7 Landau–Ginzburg theories at central charge $(X, \operatorname{vol})$8. Mathematically, it unifies:

• Derived symplectic geometry of critical loci,
• Vertex-algebraic structures and their characters,
• BV-type quantizations and higher-genus invariants (notably Witten genus analogues).

Applications include:

• Computation of Hodge-type invariants of derived critical loci,
• Categorical mirror symmetry via functors constructed from chiral VOAs and matrix factorizations,
• Enrichment of geometric representation theory by incorporating derived moduli and singularities (Bouaziz, 2024).

7. Context within the Theory of Chiral Differential Operators

The construction of chiral differential operator VOAs on derived critical loci generalizes the classical theory of CDOs:

• The VOA of CDOs is a sheaf of conformal vertex algebras, classically constructed on smooth varieties via local models or factorization algebras and classified up to $(X, \operatorname{vol})$9.
• In the singular or derived setting, dg generalizations—exemplified in the chiral critical locus construction—allow for Morita equivalence to dg algebras of differential operators on singularities, relate to the category of $f \in \mathcal{O}_X$0-modules, and connect to representation-theoretic and physical phenomena in singular moduli problems (Malikov et al., 2014, Gorbounov et al., 2016).
• The appearance of topological VOA structures and explicit formulas for characters further situates the theory at the interface between modern algebraic geometry, homological mirror symmetry, and non-perturbative aspects of supersymmetric quantum field theory.

References:

• "The chiral critical locus and topological structures" (Bouaziz, 2024)
• "Chiral De Rham complex over locally complete intersections" (Malikov et al., 2014)
• "Chiral differential operators via Batalin-Vilkovisky quantization" (Gorbounov et al., 2016)