SUMMING OVER INEQUIVALENT MAPS IN THE STRING THEORY INTERPRETATION OF2-DIMENSIONAL QCD

Following some recent work by Gross, we consider the partition functio n for QCD on a two-dimensional torus and study its stringiness. We pre sent strong evidence that the free energy corresponds to a sum over br anched surfaces with small handles mapped into the target space. The s um is modded out by all diffeomorphisms on the world sheet. This leave s a sum over disconnected classes of maps. We prove that the free ener gy gives a consistent result for all smooth maps of the torus into the torus which cover the target space p times, where p is prime, and con jecture that this is true for all coverings. Each class can also conta in integrations over the positions of branch points and small handles which act as ''moduli'' on the surface. We show that the free energy i s consistent for any number of handles and that the first few leading terms are consistent with contributions from maps with branch points.