  We consider topological field theories that compute the
Reidemeister-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 flat
gauge field. 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 also make some comments on more general supergroup Chern-Simons
theories.
