Non-Abelian Strong-Weak Coupling Duality in (String-Derived) N=4   Supersymmetric Yang-Mills Theories
  We study strong-weak coupling duality (S-duality) in N=4 supersymmetric non-Abelian Yang-Mills theories. These theories arise naturally as the low-energy limit of four-dimensional toroidal compactifications of the heterotic string. Firstly, we define the free energy in the presence of electric and magnetic fluxes using 't Hooft's prescription, i.e. through functional integrals at finite volume in the presence of twisted boundary conditions. Then, we compute those free energies in two limiting cases: small and large coupling constant. Finally, we extend the free-energies to all values of the coupling constant (and the theta angle) by presenting a fully S-duality invariant ansatz. This ansatz obeys all relevant consistency conditions; in particular, it obeys 't Hooft duality equations and Witten's magnetic-electric transmutation. The existence of an S-duality invariant, consistent definition of free energies supports the claim made in the literature that S-duality is a duality symmetry of N=4 SUSY Yang-Mills. Our ansatz also suggests that N=4, irrespective of whether partially broken or not, is in a self-dual phase: no phase transitions occur between the strong and weak coupling regimes. 
L. Girardello, A. Giveon, M. Porrati and A. Zaffaroni