Generalized modular forms

Abstract

The theory of “generalized modular forms,” initiated here, grows naturally out of questions inherent in rational conformal field theory. The latter physical theory studies q-series arising as trace functions (or partition functions), which generate a finite-dimensional SL(2, Z)-module. It is a natural step to investigate whether these q-series are in fact modular forms in the classical sense. As it turns out, the existence of the module does not, of itself, guarantee that this is so. Indeed, our Theorem 1 shows that such q-series of necessity behave like modular forms in every respect, with the important exception that the multiplier system need not be of absolute value one. The Supplement to Theorem 1 shows that such q-series are classical modular forms exactly when the scalars relating the q-series generators of the module have absolute value one. That is, the SL(2, Z)-module in question is unitary. (There is the further restriction that the associated representation is monomial.) We prove as well that there exist generalized modular forms which are not classical modular forms. (Hence, as asserted above, the q-series need not be classical modular forms.) Beyond Theorem 1 and its Supplement, which serve to relate our generalized modular forms to classical modular forms (and thus justify the name), this work develops a number of their fundamental properties. Among these are a basic result relating generalized modular forms to classical modular forms of weight 2 and so, as well, to abelian integrals. Further, we prove two general existence results and a complete characterization of weight k generalized modular forms in terms of generalized modular forms of weight 0 and classical modular forms of weight k.