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

The results obtained by Seiberg and Witten for the low-energy Wilsonianeffective actions of N=2 supersymmetric theories with gauge group SU(2) are inagreement with instanton computations carried out for winding numbers one andtwo. This suggests that the instanton saddle point saturates thenon-perturbative contribution to the functional integral. A natural frameworkin which corrections to this approximation are absent is given by thetopological field theory built out of the N=2 Super Yang-Mills theory. Afterextending the standard construction of the Topological Yang-Mills theory toencompass the case of a non-vanishing vacuum expectation value for the scalarfield, a BRST transformation is defined (as a supersymmetry plus a gaugevariation), which on the instanton moduli space is the exterior derivative. Thetopological field theory approach makes the so-called "constrained instanton"configurations and the instanton measure arise in a natural way. As aconsequence, instanton-dominated Green's functions in N=2 Super Yang-Mills canbe equivalently computed either using the constrained instanton method ormaking reference to the topological twisted version of the theory. Weexplicitly compute the instanton measure and the contribution to $u=<\Tr\phi^2>$ for winding numbers one and two. We then show that eachnon-perturbative contribution to the N=2 low-energy effective action can bewritten as the integral of a total derivative of a function of the instantonmoduli. Only instanton configurations of zero conformal size contribute to thisresult. Finally, the 8k-dimensional instanton moduli space is built using thehyperkahler quotient procedure, which clarifies the geometrical meaning of ourapproach.Comment: latex, 66 page