On the unfolding of the fundamental region in integrals of modular invariant amplitudes

We study generic one-loop (string) amplitudes where an integration over thefundamental region F of the modular group is needed. We show how the knownlattice-reduction technique used to unfold F to a more suitable region S can bemodified to rearrange generic modular invariant amplitudes. The main aim is tounfold F to the strip and, at the same time, to simplify the form of theintegrand when it is a sum over a finite number of terms, like in one-loopamplitudes for closed strings compactified on orbifolds. We give a generalformula and a recipe to compute modular invariant amplitudes. As an applicationof the technique we compute the one-loop vacuum energy \rho_n for a generic\Z_n freely acting orbifold, generalizing the result that this energy is lessthan zero and drives the system to a tachyonic divergence, and that\rho_n<\rho_m if n>m.Comment: 10 pages, 2 figure