Instanton calculus, topological field theories and N=2 Super Yang-Mills theories

The results obtained by Seiberg and Witten for N = 2 supersymmetric theorie s with gauge group SU(2) are in agreement with instanton computations carri ed out for winding numbers one and two. This suggests that the instanton sa ddle point saturates the non-perturbative contribution to the functional in tegral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N = 2 Supe r Yang-Mills theory. After extending the standard construction of the Topol ogical Yang-Mills theory to encompass the case of a non-vanishing vacuum ex pectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli spac e is the exterior derivative. We then show that each non-perturbative contr ibution to the N = 2 low-energy effective action can be written as the inte gral of a total derivative of a function of the instanton moduli. Only inst anton configurations of zero conformal size contribute to this result.