Abstract
We show that the U (1,1) (super) Chern-Simons theory is one-loop exact. This provides a direct proof of the relation between the Alexander polynomial and analytic and Reidemeister torsions. We then compute explicitly the torsions of Lens spaces and Seifert manifolds using surgery and the S and T matrices of the U (1,1) Wess-Zumino-Witten model recently determined, with complete agreement with known results. U (1,1) quantum field theories and the Alexander polynomial thus provide “toy” models with a non-trivial topological content, where all ideas put forward by Witten for SU (2) and the Jones polynomial can be explicitly checked, at finite k. Some simple but presumably generic aspects of non-compact groups, like the modified relation between Chern-Simons and Wess-Zumino-Witten theories, are also illustrated. We comment on the closely related case of GL (1,1).
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