On the instantons and the hypermultiplet mass of N=2* super Yang-Mills on S4

We show that the physical N = 4 super Yang-Mills theory on a four-sphere with an arbitrary gauge group receives no instanton contributions, by clarifying the relation between the hypermultiplet mass and the equivariant parameters of the mass-deformed theory preserving N = 2 supersymmetry. The correct relation also implies that N = 4 super conformal Yang-Mills theory with gauge group SU(2) corresponds to Liouville theory OH a torus with the insertion of a non-trivial operator, rather than the identity as have been claimed in the literature.