  Assuming the standard framework of mirror symmetry, a conjecture is
formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y
should act by families of Fourier-Mukai transforms over the complex moduli
space of the mirror X. The conjecture generalizes a proposal of Kontsevich
relating monodromy transformations and self-equivalences. Supporting evidence
is given in the case of elliptic curves, lattice-polarized K3 surfaces and
Calabi-Yau threefolds. A relation to the global Torelli problem is discussed.
