Jacobi trace functions in the theory of vertex operator algebras

We describe a type of $n$-point function associated to strongly regular vertex operator algebras $V$ and their irreducible modules. Transformation laws with respect to the Jacobi group are developed for 1-point functions. For certain elements in $V$, the finite-dimensional space spanned by the 1-point functions for the irreducible modules is shown to be a vector-valued weak Jacobi form. A decomposition of 1-point functions for general elements is proved, and shows that such functions are typically quasi-Jacobi forms. Zhu-type recursion formulas are provided; they show how an $n$-point function can be written as a linear combination of $(n-1)$-point functions with coefficients that are quasi-Jacobi forms.

2010 Mathematics Subject Classification

11F50, 17B69

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Published 12 June 2015