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

M. Trapletti
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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.

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M. Trapletti. 2002-11-28. On the unfolding of the fundamental region in integrals of modular invariant amplitudes. https://doi.org/10.1088/1126-6708%2F2003%2F02%2F012

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