  We associate to any holomorphic vertex algebra a collection of
Teichm\"{u}ller 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.

Algebraic Geometry;High Energy Physics - Theory;Representation Theory