lens spaces, with the combinatorially defined Reidemeister-Franz torsion. Finally Cheeger and Müller \[5,6\] independently proved that the analytic Ray-Singer torsion coincides with the combinatorial Reidemeister-Franz torsion in all cases.

Infinite dimensional determinants also occur naturally in quantum field theories when computing correlation functions and partition functions. In 1978 Schwarz \[7\] showed how to construct a quantum field theory on a manifold $M$ whose partition function is a power of the Ray-Singer torsion on $M$.

Schwarz’s construction uses an Abelian gauge theory but in three dimensions a non-Abelian gauge theory—the $SU(2)$ Chern-Simons theory—can be constructed and has deep and important properties established by Witten in 1988: Its partition function is the Witten invariant for the three manifold $M$ and the correlation functions of Wilson loops give the Jones polynomial invariant for the link determined by the Wilson loops—cf. \[8,9\]. Finally the weak coupling limit of the partition function is a power of the Ray-Singer torsion.

To define the Ray-Singer torsion, or simply torsion, we take a closed compact Riemannian manifold $M$ over which we have a flat bundle $E$. Let $M$ have a non-trivial fundamental group $\pi_1(M)$ which is represented on $E$—this latter property arises very naturally in the physical gauge 