SL(2,Z)-Invariant Spaces Spanned by Modular Units

Characters of rational vertex operator algebras (RVOAs) arising in 2-dimensional conformal field theories often belong (after suitable normalization) to the (multiplicative) semigroup E^+ of modular units whose Fourier expansions are in 1+q Z_{>=0}[[q]], up to a fractional power of q. If even all characters of a RVOA share this property then we have an example of what we call modular sets, i.e. finite subsets of E^+ whose elements (additively) span a vector space which is invariant under the usual action of SL(2,Z). The classification of modular sets and RVOAs seem to be closely related. In this article we give an explicit description of the group of modular units generated by E^+, we prove a certain finiteness result for modular sets contained in a natural semi-subgroup E_* of E^+, and we discuss consequences.