Chiral observables and S-duality in N = 2⋆ U(N) gauge theories

We study N 1= 2* theories with gauge group U(N) and use equivariant local- ization to calculate the quantum expectation values of the simplest chiral ring elements. These are expressed as an expansion in the mass of the adjoint hypermultiplet, with co- effcients given by quasi-modular forms of the S-duality group. Under the action of this group, we construct combinations of chiral ring elements that transform as modular forms of defnite weight. As an independent check, we con rm these results by comparing the spectral curves of the associated Hitchin system and the elliptic Calogero-Moser system. We also propose an exact and compact expression for the 1-instanton contribution to the expectation value of the chiral ring elements.