Vertex algebras and Teichmüller modular forms

Abstract

We associate to any holomorphic vertex algebra a collection of Teichmüller modular forms, one in each genus. In genus one we obtain the character of the vertex algebra, and we thus reprove Zhu's modularity result. In higher genus, we prove that these forms have an expansion in terms of the correlation functions of the vertex algebra. We propose applications to the Schottky problem, to the study of the slope of the effective cone of the moduli space of curves, and to the classification of holomorphic vertex algebras. In particular, we prove a uniqueness result for high genera partition functions of the moonshine vertex algebra.