The renormalization group equation in N=2 supersymmetric gauge theories

We clarify the mass dependence of the effective prepotential in N = 2 supersymmetric SU(N-c) gauge theories with an arbitrary number N-f < 2N(c) of flavors, The resulting differential equation for the prepotential extends the equations obtained previously for SU(2) and for zero masses, It can be viewed as an exact renormalization group equation for the prepotential, with the beta function given by a modular form. We derive an explicit formula for this modular form when N-f = 0, and verify the equation to 2-instanton order in the weak-coupling regime for arbitrary N-f and N-c. We also extend the renormalization group equation to the case of other classical gauge groups. (C) 1997 Elsevier Science B.V.