Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras

TL;DR

This paper investigates the modular properties of trace functions in free boson and lattice vertex operator algebras, revealing their quasi-modular and modular nature respectively, and explores the connection to spherical harmonics.

Contribution

It demonstrates that trace functions in these VOAs are quasi-modular or modular and links spherical harmonics to primary fields in lattice theories.

Findings

Trace functions are of the form f(q)/η(q)^d with f(q) quasi-modular or modular.

In free boson VOAs, trace functions are quasi-modular forms.

In lattice VOAs, trace functions are sums of holomorphic modular forms.

Abstract

We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/\eta(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.