Low-energy spectrum of Script N = 4 super-Yang-Mills onT3: flat connections, bound states at threshold, andS-duality

We study (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatialthree-torus. The low energy spectrum consists of a number of continua of statesof arbitrarily low energies. Although the theory has no mass-gap, it appearsthat the dimensions and discrete abelian magnetic and electric 't Hooft fluxesof the continua are computable in a semi-classical approximation. Thewave-functions of the low-energy states are supported on submanifolds of themoduli space of flat connections, at which various subgroups of the gauge groupare left unbroken. The field theory degrees of freedom transverse to such asubmanifold are approximated by supersymmetric matrix quantum mechanics with 16supercharges, based on the semi-simple part of this unbroken group. Conjecturesabout the number of normalizable bound states at threshold in the latter theoryplay a crucial role in our analysis. In this way, we compute the low-energyspectra in the cases where the simply connected cover of the gauge group isgiven by SU(n), Spin(2n+1) or Sp(2n). We then show that the constraints ofS-duality are obeyed for unique values of the number of bound states in thematrix quantum mechanics. In the cases based on Spin(2n+1) and Sp(2n), theproof involves surprisingly subtle combinatorial identities, which hint at arich underlying structure.Comment: 28 pages. v2:reference adde