Abstract
The three-dimensional Chern-Simons gauge theory is a top ological quantum eld theory, whose correlation functions give metric-indep endent invariants of knots and three-manifolds. In this thesis, we consider a version of this theory, in which the gauge group is taken to b e a Lie sup ergroup. We show that the analytically-continued ver- sion of the sup ergroup Chern-Simons theory can b e obtained by top ological twisting from the low energy eective theory of the intersection of D3- and NS5-branes in the typ e I IB string theory. By S -duality, we deduce a dual magnetic description; and a slightly dierent duality, in the case of orthosymplectic gauge group, leads to a strong-weak coupling duality b etween certain sup ergroup 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 op erators. We also consider the purely three-dimensional version of the psl (1 j 1) and the U (1 j 1) sup ergroup Chern-Simons, coupled to a background complex at gauge eld. These theories compute the Reidemeister-Milnor-Turaev torsion in three dimensions. We use the 3d mirror symmetry to derive the Meng-Taub es theorem, which relates the torsion and the Seib erg-Witten invariants, for a three-manifold with arbitrary rst Betti numb er. We also present the Hamiltonian quantization of our theories, nd the mo dular transformations of states, and various prop erties of lo op op erators. Our results for the U(1|1) theory are in general consistent with the results, found for the GL(1|1) WZW mo del. We exp ect our ndings to b e useful for the construc- tion of Chern-Simons invariants of knots and three-manifolds for more general Lie sup ergroups.