Reidemeister torsion, the Alexander polynomial and $U(1,1)$ Chern-Simons theory

Details Reidemeister torsion, the Alexander polynomial and $U(1,1)$ Chern-Simons theory Reidemeister torsion, the Alexander polynomial and $U(1,1)$ Chern-Simons theory

1992-09-21

arxiv.org/abs/hep-th/9209073v1

J.Geom.Phys. 13 (1994) 105-123

Physics

High Energy Physics

High Energy Physics - Theory

21 pages, no figures

Scientific paper

10.1016/0393-0440(94)90022-1

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 torsion. We then proceed to compute explicitely 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 provide thus "toy" models with a non trivial topological content, where all ideas put forward by Witten for $SU(2)$ and the Jones polynomial can be explicitely 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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