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We consider topological field theories that compute theReidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1)and the U(1|1) Chern-Simons theories, coupled to a background complex flatgauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem,which relates the torsion and the Seiberg-Witten invariants, for athree-manifold with arbitrary first Betti number. We also present theHamiltonian quantization of our theories, find the modular transformations ofstates, 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) WZWmodel. We also make some comments on more general supergroup Chern-Simonstheories.

Analytic Torsion, 3d Mirror Symmetry And Supergroup Chern-Simons Theories