Gauge theory duality and Chern--Simons theory

A novel relationship between four-dimensional supersymmetric gauge theories and three-dimensional Chern-Simons theories is explored. We conjecture that when the gauge theory is self-dual under the electric-magnetic duality, the unitary operator that realizes the duality defines a Chern-Simons theory under certain restrictions. Equivalently, we argue that when the four-dimensional theory is compactified on a circle with duality twist, the low energy limit is described by a three-dimensional topological theory, which we conjecture to be a Chern-Simons theory.;The conjecture is tested by computing the number of ground states in two different settings. When the theory is further compactified on a torus, the number of ground states of the compactified gauge theory exactly matches the number of states of the Chern-Simons theory. Moreover, modular invariance of the Chern-Simons theory on the torus can be related to T-duality of string theory in this setting. When the theory is compactified on a Riemann surface of genus two, we can match the results only up to several sign ambiguities.