Partition function of N=2^∗ SYM on a large four-sphere

We examine the partition function of N=2^∗ supersymmetric SU(N) Yang-Mills theory on the four-sphere in the large radius limit. We point out that the large radius partition function, at fixed N, is computed by saddle-points lying on walls of marginal stability on the Coulomb branch of the theory on ℝ⁴. For N an even (odd) integer and θ_YM=0 (π), these include a point of maximal degeneration of the Donagi-Witten curve to a torus where BPS dyons with electric charge [(N/2)] become massless. We argue that the dyon singularity is the lone saddle-point in the SU(2) theory, while for SU(N) with N>2, we characterize potentially competing saddle-points by obtaining the relations between the Seiberg-Witten periods at such points. Using Nekrasov's instanton partition function, we solve for the maximally degenerate saddle-point and obtain its free energy as a function of g_YM and N, and show that the results are "large-N exact". In the large-N theory our results provide analytical expressions for the periods/eigenvalues at the maximally degenerate saddle-point, precisely matching previously known formulae following from the correspondence between N=2^∗ theory and the elliptic Calogero-Moser integrable model. The maximally singular point ceases to be a saddle-point of the partition function above a critical value of the coupling, in agreement with the recent findings of Russo and Zarembo.