Discrete Symmetries of Functional Determinants

We study discrete (duality) symmetries of functional determinants. An exact transformation of the effective action under the inversion of background fields $\beta (x)\to {\beta }^{-1}(x)$ is found. We show that in many cases this inversion does not change functional determinants. Explicitly studied models include a matrix theory in two dimensions, the dilaton-Maxwell theory in four dimensions on manifolds without a boundary, and a two-dimensional dilaton theory on manifolds with boundaries. Our results provide an exact relation between strong and weak coupling regimes with possible applications to string theory, black hole physics and dimensionally reduced models.