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Cohomology of Jacobi forms

A. Zuevsky

TL;DR

This work constructs a cohomology theory for Jacobi forms arising from vertex operator algebras by developing reduction formulas that express $n$-point Jacobi functions in terms of lower-point data. It defines a chain complex of Jacobi $n$-point functions on the torus ${\mathcal T}$ with a coboundary operator built from reduction data, and identifies a domain ${\mathfrak V}_n$ where the chain condition holds. The main result links the $n$-th reduction cohomology $H^n_J(W)$ to analytic continuations of solutions to Knizhnik–Zamolodchikov–type equations, giving a geometric interpretation as multipoint connections on a deformed VOA bundle over the torus. The framework encompasses orbifold and shifted-Virasoro cases, and the appendices provide background on quasi-Jacobi forms, deformed elliptic functions, and square bracket formalisms essential to the construction. Overall, the paper connects non-commutative VOA structures with modular-analytic and geometric aspects of Jacobi forms, offering computational tools and conceptual links to KZ theory and Bott–Segal-type theorems.

Abstract

We define and compute a cohomology of the space of Jacobi forms based on precise analogues of Zhu reduction formulas. A counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi forms on the torus is proven. It is shown that the reduction cohomology for Jacobi forms is given by the cohomology of $n$-point connections over a deformed vertex operator algebra bundle defined on the torus. The reduction cohomology for Jacobi forms for a vertex operator algebra is determined in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations.

Cohomology of Jacobi forms

TL;DR

This work constructs a cohomology theory for Jacobi forms arising from vertex operator algebras by developing reduction formulas that express -point Jacobi functions in terms of lower-point data. It defines a chain complex of Jacobi -point functions on the torus with a coboundary operator built from reduction data, and identifies a domain where the chain condition holds. The main result links the -th reduction cohomology to analytic continuations of solutions to Knizhnik–Zamolodchikov–type equations, giving a geometric interpretation as multipoint connections on a deformed VOA bundle over the torus. The framework encompasses orbifold and shifted-Virasoro cases, and the appendices provide background on quasi-Jacobi forms, deformed elliptic functions, and square bracket formalisms essential to the construction. Overall, the paper connects non-commutative VOA structures with modular-analytic and geometric aspects of Jacobi forms, offering computational tools and conceptual links to KZ theory and Bott–Segal-type theorems.

Abstract

We define and compute a cohomology of the space of Jacobi forms based on precise analogues of Zhu reduction formulas. A counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi forms on the torus is proven. It is shown that the reduction cohomology for Jacobi forms is given by the cohomology of -point connections over a deformed vertex operator algebra bundle defined on the torus. The reduction cohomology for Jacobi forms for a vertex operator algebra is determined in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations.

Paper Structure

This paper contains 20 sections, 11 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Chain complex for vertex operator algebra $n$-point functions
  3. Spaces of $n$-point Jacobi functions via vertex operators
  4. Reduction formulas and examples of coboundary operators for Jacobi $n$-point functions
  5. The coboundary operator
  6. The simplest coboundary operator
  7. Coboundary operator for a shifted Virasoro vector
  8. Vertex operator superalgebra case
  9. Cohomology
  10. The $n$-th cohomology and analytic extensions of solutions to Knizhnik-Zamolodchikov equations
  11. Geometrical meaning of reduction formulas and conditions \ref{['conditions']}
  12. Appendix: Quasi--Jacobi forms
  13. Modular and elliptic functions
  14. Deformed elliptic functions
  15. Appendix: Reduction formulas for Jacobi $n$-point functions
  16. ...and 5 more sections

Key Result

Proposition 1

In notations and under assumptions of Subsections main--vosa, the $n$-th reduction cohomology of the space of Jacobi forms for a $V$-module $W$ is given by space of analytical continuations of solutions $\mathcal{Z}_{W}^J \left({\bf x}_n; B \right)$ to the equation with $x_i \notin {\mathfrak V}_{i}$, $\beta=h$ for a shifted Virasoro element and zero otherwise, for $1 \le i \le n$. These are give

Theorems & Definitions (18)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Lemma 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Proposition 2
  • ...and 8 more