  In topological field theories determinants of maps with negative as well as
positive eigenvalues arise. We give a generalisation of the zeta-regularisation
technique to derive expressions for the phase and scaling-dependence of these
determinants. For theories on odd-dimensional manifolds a simple formula for
the scaling dependence is obtained in terms of the dimensions of certain
cohomology spaces. This enables a non-perturbative feature of Chern-Simons
gauge theory to be reproduced by path-integral methods.
