theory context where it corresponds simply to the space of flat connections all of whose content resides in their holonomy—In any case the torsion is then the real number $T(M,E)$ where $$\ln T(M,E)=\sum_0^n(-1)^qq\ln \Det \Delta^E_q,\quad n=\dim M\no$$ The metric independence of the torsion requires that we assume, in the above definition, that the cohomology ring $H^*(M;E)$ is trivial; this means that the Laplacians $\Delta^E_q$ have empty kernels and so are strictly positive definite. Given this fact one may use zeta functions to define $ \Det \Delta^E_q$ in the standard way. Recall that if $P$ is a positive elliptic differential or pseudo-differential operator with spectrum $\{\mu_n\}$ and degeneracies $\{\Gamma_n\}$ then its associated zeta function $\zeta_P(s)$ is a meromorphic function of $s$, regular at $s=0$, which is given by $$\zeta_P(s)=\sum_{\mu_n}{\Gamma_n\over\mu_n^s}\no$$ and its determinant $\Det  P$ is defined by $$\ln\Det  P=-\left.{d\zeta_P(s)\over ds}\right\vert_{s=0}\no$$ Using this we have $$\ln T(M,E)=
-\sum_0^n(-1)^qq\left.{d\zeta_{\Delta_q^{E}}(s)\over ds}\right\vert_{s=0}\no$$

Quantum field theories of the type alluded to above are usually referred to as topological quantum field theories or simply topological field theories.

It turns out that more than one topological field theory can be used to give the torsion, for an excellent review of this question cf. Birmingham et al. \[10\]. For 