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

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 $\mathbf f{\boldsymbol(\mathbf q\boldsymbol)}\boldsymbol/\mathbf\eta{\boldsymbol(\mathbf q\boldsymbol)}^\mathbf d$ 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.