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

TL;DR

This paper demonstrates that $U(1,1)$ Chern-Simons theory is one loop exact, establishing a direct link between the Alexander polynomial, torsion invariants, and topological quantum field theories, with explicit calculations for specific manifolds.

Contribution

It provides the first explicit computations of torsions for Lens spaces and Seifert manifolds using $U(1,1)$ WZW model data, illustrating the topological content of non-compact gauge theories.

Findings

$U(1,1)$ Chern-Simons theory is one loop exact.

Explicit torsion calculations match known results.

$U(1,1)$ models serve as tractable examples for topological quantum field theories.

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 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 $GL(1,1)$.