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

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

2002-11-28

arxiv.org/abs/hep-th/0211281v1

JHEP 0302 (2003) 012

Physics

High Energy Physics

High Energy Physics - Theory

10 pages, 2 figures

Scientific paper

10.1088/1126-6708/2003/02/012

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 \rho_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 \rho_n<\rho_m if n>m.

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