  We study the dynamics of supersymmetric theories in five dimensions obtained
by compactifications of M-theory on a Calabi-Yau threefold X. For a compact X,
this is determined by the geometry of X, in particular the Kahler class
dependence of the volume of X determines the effective couplings of vector
multiplets. Rigid supersymmetry emerges in the limit of divergent volume,
prompting the study of the structure of Duistermaat-Heckman formula and its
generalizations for non-compact toric Kahler manifolds. Our main tool is the
set of finite-difference equations obeyed by equivariant volumes and their
quantum versions. We also discuss a physical application of these equations in
the context of seven-dimensional gauge theories, extending and clarifying our
previous results. The appendix by M. Vergne provides an alternative local proof
of the shift equation.
