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
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