Cohomology of Jacobi forms

TL;DR

This paper introduces a cohomology theory for Jacobi forms, linking it to vertex operator algebras and KZ equations, and proves a Bott-Segal type theorem for this setting.

Contribution

It defines and computes a new cohomology for Jacobi forms, establishing analogues of classical theorems and connecting to vertex operator algebra structures.

Findings

Reduction cohomology is given by the cohomology of n-point connections.

Proves a Bott-Segal type theorem for Jacobi forms.

Determines cohomology in terms of solutions to KZ equations.

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.