S-duality and a large N phase transition in N = 4 on K3 at strong coupling

We study the supersymmetric partition function of N = 4 super Yang-Mills with gauge group SU(N) on K3 in the large N , fixed g limit and show that it undergoes a first order phase transition at the S-duality invariant value of the gauge coupling g. Turning on the θ-angle we find lines of phase transitions on the τ plane. The resulting phase diagram and the large N free energy are exactly SL(2, Z) invariant. Similar phase transitions take place in systems related to the N = 4 on K3 by dualities. One of them is the Dabholkar-Harvey heterotic string system. We consider its mixed (a la Ooguri-StromingerVafa) partition function allowing contributions from multi-string states. We find that in the large winding charge limit, it undergoes a phase transition with respect to chemical potential for momentum. It is a short-string, long-string transition that we find interesting in connection with black hole entropy counting.