Cohomology of Jacobi forms

Technical Summary: Cohomology of Jacobi Forms

Problem Statement
The paper addresses the computation of continuous cohomologies for non-commutative structures on manifolds, specifically within the context of Vertex Operator Algebras (VOAs). While classical cohomology of holomorphic vector fields on Riemann surfaces is often trivial, the author seeks to develop a cohomology theory for the space of Jacobi $n$-point functions generated by a VOA. These functions, which include an elliptic variable to track a "charge" via an element $J \in V_1$, satisfy functional equations known as Zhu reduction formulas. The central problem is to construct a cochain complex where these reduction formulas serve as coboundary operators, thereby defining a "reduction cohomology" that encodes the non-commutative structure of the VOA and yields functional equations for quasi-modular forms.

Methodology
The author constructs a cochain complex $(C^\bullet(W), \delta^\bullet)$ where:

1. Cochain Spaces ($C^n(W)$): Defined as the space of Jacobi $n$-point functions $Z^J_W(x_n; B)$ associated with a $V$-module $W$, where $x_n$ represents a tuple of vectors and local coordinates on a torus $T$, and $B = (\tau, z)$ represents the modular and elliptic parameters.
2. Coboundary Operators ($\delta_n$): Defined explicitly using Zhu-type reduction formulas (specifically those established by Bringmann-Krauel-Tuite for Jacobi forms). The operator $\delta_n$ maps an $n$-point function to an $(n+1)$-point function via a sum involving meromorphic coefficient functions $f_{k,m}$ and the insertion of square-bracket modes $v_{n+1}[m]_\beta$ into the $k$-th slot of the input.
3. Chain Condition: The condition $\delta_{n+1} \circ \delta_n = 0$ is imposed, which translates into a system of functional-differential equations (analogous to the Knizhnik-Zamolodchikov equations) that the $n$-point functions must satisfy.
4. Cases Analyzed: The explicit form of the coboundary operator is derived for four specific scenarios:
   • The general case.
   • The simplest case.
   • The shifted-Virasoro case (involving a shifted Virasoro element).
   • The vertex operator superalgebra/orbifold case.

Key Contributions and Results
The paper establishes two primary theorems characterizing the resulting cohomology groups $H^n_J(W)$:

1. KZ Characterization (Theorem 4.1):
   The $n$-th reduction cohomology $H^n_J(W)$ is isomorphic, as a complex vector space, to the space of analytic continuations of non-zero solutions to a specific equation:
   $$\sum_{k=0}^n \sum_{m \ge 0} f_{k,m}(x_n; B) T_k(v_{n+1}[m]_\beta) Z^J_W(x_n; B) = 0$$
   This equation is identified as a Jacobi-form analogue of the Knizhnik-Zamolodchikov (KZ) equations. The solutions are expressible as series of deformed Weierstrass functions, recursively generated by the reduction formulas.

2. Bott-Segal Type Theorem (Lemma 4.6):
   The paper demonstrates that Jacobi $n$-point functions generated by the reduction formulas act as $n$-point connections on the space of $g$-automorphism-deformed sections of the VOA bundle over the torus. Consequently, the reduction cohomology is isomorphic to the cohomology of the space of these deformed sections:
   $$H^n_J(W) \cong \text{Conn}_n / G_{n-1}$$
   This result provides a geometric interpretation of the algebraic reduction formulas, linking them to the theory of connections on bundles, analogous to results by Bott, Segal, and Wagemann for holomorphic vector fields.

Additionally, the paper notes that for the orbifold case, the cohomology contains functions expressible as determinants of matrices with entries being deformed elliptic functions.

Significance and Claims
The author positions this work as a foundational step toward a complete description of the cohomologies of holomorphic objects arising from non-commutative structures on complex manifolds. The significance lies in:

• Unification: Simultaneously encoding the non-commutative structure of the VOA, producing functional equations for quasi-modular forms, and admitting a geometric interpretation via connections on a VOA bundle.
• Generalization: Operating at the level of $n$-point functions for all $n \ge 0$ simultaneously, incorporating the extra Jacobi (elliptic) variable, which distinguishes it from previous work on first and second cohomologies of grading-restricted vertex algebras (e.g., Huang).
• Applicability: The theory is presented as extendable to higher-genus Riemann surfaces and applicable to the study of integrable models and deformation theory.

The paper explicitly states that no datasets were generated or analyzed, and the results are purely theoretical derivations based on existing VOA axioms and reduction formulas.