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 f ields beta (x) --> beta (-1)(x) is found. We show that in many cases this i nversion does not change functional determinants. Explicitly studied models include a matrix theory in two dimensions, the dilaton Maxwell theory in f our dimensions on manifolds without a boundary, and a two-dimensional dilat on theory on manifolds with boundaries. Our results provide an exact relati on between strong and weak coupling regimes with possible applications to s tring theory, black hole physics and dimensionally reduced models. (C) 2001 Elsevier Science B.V, All rights reserved.