Nonperturbative relations in N=2 supersymmetric Yang-Mills theory and the Witten-Dijkgraaf-Verlinde-Verlinde equation

We find the nonperturbative relation between [tr phi(2)], [tr phi(3)] the prepotential F and [phi(i)] in N = 2 supersymmetric Yang-Mills theory (SYM) with gauge group SU(3). Nonlinear differential equations for F including the Witten-Dijkgraaf-Verlinde-Verlinde equation are obtained, indicating that N = 2 SYM theories are essentially topological field theories which should be seen as the low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods, and find the beta function in terms of the moduli. In doing this we discuss the uniformization problem for the quantum moduli space.