Vertex operator algebra bundles on modular curves and their associated modular forms

This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to $V$-valued quasi-modular forms. The space $Q(V)$ of $V$-valued quasi-modular forms is endowed with the structure of a doubled QVOA, and in particular the algebra $Q$ of quasi-modular forms is itself a doubled QVOA. $Q(V)$ also admits a natural derivative operator arising from the connection on the bundle defined by $V$ and the modular derivative, which we call the raising operator. We introduce an associated lowering operator $\Lambda$ on $Q(V)$ having the property that the $V$-valued modular forms $M(V)\subseteq Q(V)$ are the kernel of $\Lambda$. This extends the classical theory of scalar-valued quasi-modular forms. We exhibit an explicit isomorphism of $M(V)$ with $M \otimes V$. Finally, the coordinate invariance of vertex operators implies that $M(V)$ has a natural Hecke theory, and we use this isomorphism to fully describe the Hecke eigensystems: they are the same as the systems of eigenvalues that arise from scalar-valued quasi-modular forms.