Vertex Algebra of Chiral Differential Operators

• Vertex algebra of chiral differential operators is a framework that constructs sheaves of vertex algebras from smooth vector bundles equipped with connections.
• It integrates differential geometry, representation theory, and quantum field theory by encoding geometric data and symmetries through souped-up Lie algebras and vertex functors.
• The construction generalizes the chiral de Rham complex to nontrivial bundles, enabling applications in quantum cohomology, anomaly analysis, and mirror symmetry.

The vertex algebra of chiral differential operators (CDOs) provides a formalism that unifies differential geometry, representation theory, and the algebraic structures of quantum field theory within the framework of vertex algebras. Central to this realization is the notion that, starting from smooth vector bundles (typically equipped with connections and group actions), one can construct sheaves of vertex algebras encapsulating both the local geometry and the symmetries underlying classical and quantum aspects of fields on manifolds. This construction generalizes the chiral de Rham complex, extending it to genuinely nontrivial bundles and producing new structures with deep applications in geometry and physics.

1. Algebraic and Geometric Construction

Given a smooth $G$-vector bundle $E \to M$ with connection $\nabla$, the foundational object is the sheaf of vertex algebras $\mathcal{E}^{ch(E, \nabla)}$. The construction proceeds as follows:

• Souped-up Lie Algebra: Classical data are captured via

$$\mathcal{E}^{(E, \nabla)} = \mathfrak{s}(\mathcal{A} \otimes (\Gamma(SE \otimes \Lambda E)) \otimes \mathfrak{sX}_{(\nabla)}, \mathcal{A} \otimes \Gamma(SE \otimes \Lambda E))$$

where $\mathcal{A}$ is typically the exterior algebra of differential forms $\Omega$, $\Gamma(SE \otimes \Lambda E)$ denotes sections of (symmetric)$\otimes$(antisymmetric) tensor bundles, and $\mathfrak{sX}_{(\nabla)}$ is a Lie algebra generated by: - The connection $\nabla$ acting as a derivation, - Infinitesimal gauge transformations (endomorphisms of $E$), - Contraction operators $\iota_X$ for $X$ a vector field.

• Vertex Algebra Functor: Applying a functor $\mathcal{V}$ yields

$$V(\mathcal{E}^{(E, \nabla)}) = F(\mathcal{E}^{(E, \nabla)})/K(\mathcal{E}^{(E, \nabla)})$$

with $F(...)$ the free vertex algebra generated by the souped-up Lie algebra, products $x_n$, and modes, and $K(...)$ an ideal encoding both vertex algebraic relations (identity, locality, translation, etc.) and compatibility with classical products and support (see Section 4 below).

• Role of the Connection: The connection $\nabla$ is realized as both a generator (in bracket relations such as $[\nabla, \iota_X] = \mathcal{L}_X$) and a derivation on the vertex algebra:

$$\nabla_0 (x_n y) = (\nabla_0 x)_n y + x_n (\nabla_0 y)$$

imparting a quantized, chiral avatar of the classical differential geometric connection.

2. Internal Structure: Subsheaves and Symmetries

The sheaf $\mathcal{E}^{ch(E,\nabla)}$ contains several significant subsheaves:

• Classical Subsheaf: The superalgebra $\Omega \otimes \Gamma(SE \otimes \Lambda E)$, preserved in the vertex algebra due to the design of the ideal $K$ and supporting the familiar wedge and symmetric products of forms and sections.
• Lie Algebraic Subsheaf: Generated by $\nabla$, gauge transformations, and contraction operators $\iota_X$, forming a Lie superalgebra with key structural brackets:
  • $[\nabla, \iota_X] = \mathcal{L}_X$ (Cartan-type formula),
  • $[\iota_X, \iota_Y] = 0$,
  • Infinitesimal gauge transformations act by vertical derivations on $E$-factors.

The contraction operators $\iota_X$ reflect odd (fermionic) symmetries, anticommutativity, and geometric probing along vector fields; gauge transformations model local trivializations and preserve bundle structure.

3. Connection to Chiral de Rham Sheaf

In the special case $E = M \times \mathbb{C}$ (the trivial line bundle) with $\nabla = d$, one recovers the chiral vector bundle $\mathcal{E}^{ch(M \times \mathbb{C}, d)}$:

• Embedding of de Rham Complex: The classical subalgebra $\Omega$ (forms on $M$) embeds as a vertex subalgebra, paralleling the chiral de Rham construction, with the chiralized action of $d$ producing a global field

$$d(\zeta) = \sum_{n \in \mathbb{Z}} \frac{d \circ_n}{\zeta^{n+1}}$$

without requiring further geometric restrictions (in contrast to, e.g., the original chiral de Rham sheaf, which for global fields may impose Calabi–Yau conditions).

• Enhanced Structure: In contrast to the classic chiral de Rham sheaf, additional generators ensure that the entire souped-up Lie algebra structure, including gauge and derivative data, is retained as a subsheaf, enabling this construction to serve as a genuine extension of chiral de Rham to bundles with connections.

4. Formalism and Key Equations

The construction is formalized using a combination of algebraic and sheaf-theoretic techniques:

Object	Definition/Formula	Purpose
Vertex algebra $V(\mathfrak{v})$	$F(\mathfrak{v})/I(\mathfrak{v})$	Free algebra modulo vertex algebra ideal
Lie algebra generators	$s_0 t - [s, t]$ for $s,t$ in Lie algebra	Enforce compatibility with Lie bracket in modes
Action of commutative algebra	$a_{-1} s - as$ for $a$ in commutative algebra	Match module structure
Module composition consistency	$(ab)_{-1} x - a_{-1}(b_{-1} x)$	Respect classical product
State–field correspondence	$$x(\zeta) = \sum_{n \in \mathbb{Z}} \frac{x_n}{\zeta^{n+1}}$$	Mode expansion
Vertex derivation $D$	$D = \circ_{-2} 1$	Translation operator
Connection–contraction commutator	$[\nabla, \iota_X] = \mathcal{L}_X$	Vertex Cartan formula
Support-killing generator in $K$	$k[x] := x - \pi(x)$, with $\pi$ nullifying zero-support monomials	Enforces sheaf locality

The collation of the ideal $K$ ensures that the global sections truly have local support and that the chiral vector bundle structure descends appropriately to the sheaf level.

5. Applications in Geometry and Mathematical Physics

The algebraic encoding of geometric and symmetry data within vertex algebras has several far-reaching implications:

• Chiral Differential Geometry in QFT: Vertex algebras describing operator product expansions are naturally adapted to encode geometric data, including connections and gauge symmetries, in a “chiral” context, allowing for the fusion of classical differential geometry and quantum field theoretic structure.
• Quantized Characteristic Classes and Index Theory: The chiral structure enables the computation of characteristic classes (e.g., Chern–Weil forms via $\nabla^2 = \frac{1}{2}[\nabla, \nabla]$) as elements within the chiral vertex algebra, providing a “quantized” deformation of classical index-theoretic invariants.
• Anomalies and Dualities: Built-in gauge and contraction operators facilitate the cohomological and algebraic paper of field theory anomalies and dualities via the representation theory of the vertex algebra.
• Extension to Nontrivial Bundles and Connections: The effect of nontrivial holonomy and curvature is encoded directly in the vertex algebra’s representation theory, a structure not present in the de Rham or standard chiral de Rham sheaf, and is potentially relevant in examining mirror symmetry and topological field theory scenarios.
• Functoriality and Sheaf-theoretic Connections: The passage from local (or presheaf-level) data—differential forms, sections, infinitesimal symmetries—to globally defined vertex algebra sheaves is functorial with respect to bundle morphisms, aligning with fundamental constructions in modern algebraic geometry.

6. Functorial Properties and Extensions

The methodology underlying the vertex algebra of chiral differential operators is inherently functorial: starting from the data of $\Omega^0$–modules or souped-up Lie algebras built from $E$ and $\nabla$, the procedure yields a sheaf of vertex algebras $\mathcal{E}^{ch(E,\nabla)}$. The construction is robust enough to encompass not only smooth vector bundles with connections but also allows potential generalization—through the choice of generators and relations in $K$—to singular bundles, bundles with additional structures (super or holomorphic bundles), and to the context of derived geometry.

7. Theoretical Impact and Future Directions

The synthesis of differential-geometric, representation-theoretic, and physical perspectives via the vertex algebra of chiral differential operators offers multiple research avenues:

• String Theory and CFT: The association of geometric data to chiral algebraic structures models the operator content of conformal field theories and string backgrounds equipped with gauge bundles and connections.
• Quantum Cohomology and Elliptic Genera: Embedding classical cohomological invariants (such as the Witten genus) into a chiral algebraic framework provides new approaches to the paper of higher-genus invariants and index theory in the quantized regime.
• Functorial Sheaf Theory in Complex and Algebraic Geometry: By establishing a systematic functor mapping geometric bundles (with connections) into algebraic sheaves of vertex algebras, the construction forges deep links between vertex algebra theory and classical tools in algebraic and complex geometry, opening the possibility of employing methods from homological algebra, D-modules, and sheaf cohomology in the analysis of vertex algebraic objects.
• Representation Theory and Mirror Symmetry: The intricate dependence of the chiral vertex algebra on the underlying bundle's topology and connection promises applications to the analysis of mirror pairs and the investigation of dualities in representation theory and mathematical physics.

The construction and paper of vertex algebras of chiral differential operators thus stand as a central pillar in the interface of geometric representation theory, algebraic geometry, and mathematical physics, providing a universal language for the quantization and global paper of geometric and symmetry data on manifolds.