Nonperturbative Relations in N=2 Supersymmetric Yang-Mills Theory and the Witten-Dijkgraaf-Verlinde-Verlinde Equation

We find the nonperturbative relation between left-angle trφ2 right-angle,left-angle trφ3 right-angle the prepotential F and left-angle φi right-angle 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 β function in terms of the moduli. In doing this we discuss the uniformization problem for the quantum moduli space. copyright 1996 The American Physical Society