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

Abstract

We study generic one-loop (string) amplitudes where an integration over the fundamental region F of the modular group is needed. We show how the known lattice-reduction technique used to unfold F to a more suitable region S can be modified to rearrange generic modular invariant amplitudes. The main aim is to unfold F to the strip and, at the same time, to simplify the form of the integrand when it is a sum over a finite number of terms, like in one-loop amplitudes for closed strings compactified on orbifolds. We give a general formula and a recipe to compute modular invariant amplitudes. As an application of the technique we compute the one-loop vacuum energy ρ_n for a generic \Z_n freely acting orbifold, generalizing the result that this energy is less than zero and drives the system to a tachyonic divergence, and that ρ_n<ρ_m if n>m.