Supersymmetric index in four-dimensional gauge theories

This paper is devoted to a systematic discussion of the supersymmetric index Tr (-1)^ for the minimal supersymmetric Yang-Mills theory -with any simple gauge group G -primarily in four spacetime dimensions.The index has refinements that probe confinement and oblique confinement and the possible spontaneous breaking of chiral symmetry and of global symmetries, such as charge conjugation, that are derived from outer automorphisms of the gauge group.Predictions for the index and its refinements are obtained on the basis of standard hypotheses about the infrared behavior of gauge theories.The predictions are confirmed via microscopic calculations which involve a Born-Oppenheimer computation of the spectrum as well as mathematical formulas involving triples of commuting elements of G and the Chern-Simons invariants of flat bundles on the three-torus.