UC Santa Cruz

This thesis develops a theory relating the Jacobi group with n-point functions associated with strongly regular vertex operator algebras. The n-point functions considered here have additional complex variables and generalize n-point functions studied in other works in the mathematics and physics literature. Recursion formulas are discussed which reduce the study of n-point functions to the study of 1-point and 0-point functions.

We consider the space of 1-point functions associated to inequivalent irreducible admissible modules for a strongly regular vertex operator algebra. We develop transformation laws for this space of functions under the Jacobi group. With additional assumptions, we show that 1-point functions are sums of products of 1-point functions of modules for the commutant subVOA of the vertex operator algebra together with a type of Jacobi theta series. Conditions will be given where these functions are vector-valued weak Jacobi forms. A number of corollaries to these results are developed, including a sharper result in the case of holomorphic vertex operator algebras.

Other results contained in this thesis include transformation laws for Jacobi theta functions with spherical harmonics, and a generalization of a result of Miyamoto to include zero modes of elements.