Cohomology of Jacobi forms

Abstract

We define and study a cohomology theory for the space of Jacobi $n$-point functions generated by a vertex operator (super)algebra, using precise analogues of Zhu's reduction formulas. A cochain complex $(C^{\bullet}(W), δ^{\bullet})$ is constructed whose coboundary operators are given by Zhu-type reduction maps, and whose cohomology groups $H^{n}_{J}(W)$ we call the {reduction cohomology of Jacobi forms}. We prove that the $n$-th reduction cohomology of a $V$-module $W$ is isomorphic to the space of analytic continuations of solutions to a vertex-operator-algebraic analogue of the Knizhnik-Zamolodchikov equations. We further show that Jacobi $n$-point reduction formulas are $n$-point connections on the vertex operator algebra bundle over the torus, yielding a Bott-Segal-type theorem: $H^{n}_{J}(W)$ is isomorphic to the cohomology of the space of deformed sections of the VOA bundle.