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 nonAbelian 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 find 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. String theories compactified on a circle are known to possess duality invariances: their spectrum and interactions are invariant under a transformation that maps the radius of the circle, R, into α/R (α is the string tension). This symmetry holds at each order in the string loop expansion, and, in general, interchanges light states with ultra-massive ones. It is therefore essentially non perturbative in α and it is sometimes called target-space duality. Recently another type of duality symmetry has been conjectured. It is called strong-weak coupling duality, or S-duality [1]. It is a symmetry of four-dimensional strings, which acts by transforming the four-dimensional dilaton field S as iS → aiS + b ciS + d , (