Aspects Of Supergroup Chern-Simons Theories

Abstract

The three-dimensional Chern-Simons gauge theory is a topological quantum field theory, whose correlation functions give metric-independent invariants of knots and three-manifolds. In this thesis, we consider a version of this theory, in which the gauge group is taken to be a Lie supergroup. We show that the analytically-continued version of the supergroup Chern-Simons theory can be obtained by topological twisting from the low energy effective theory of the intersection of D3- and NS5-branes in the type IIB string theory. By S-duality, we deduce a dual magnetic description; and a slightly different duality, in the case of orthosymplectic gauge group,

leads to a strong-weak coupling duality between certain supergroup Chern-Simons theories on R^3. Some cases of these statements are known in the literature. We analyze how these dualities act on line and surface operators.

We also consider the purely three-dimensional version of the psu(1|1) and the U(1|1) supergroup Chern-Simons, coupled to a background complex flat gauge field. These theories compute the Reidemeister-Milnor-Turaev torsion in three dimensions. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and the Seiberg-Witten invariants, for a three-manifold with arbitrary first Betti number. We also present the Hamiltonian quantization of our theories, find the modular transformations of states, and various properties of loop operators. Our results for the U(1|1) theory are in general consistent with the results, found for the GL(1|1) WZW model. We expect our findings to be useful for the construction of Chern-Simons invariants of knots and three-manifolds for more general Lie supergroups.