Abstract
Abstract In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus. Unlike the traditional approach, the Narain lattice does not play any role in the unfolding procedure, T-duality is kept manifest at all steps, a choice of Weyl chamber is not required and the analytic structure of the amplitude is transparent. In the present paper, we generalise this procedure to the case of Abelian $ {{\mathbb{Z}}_N} $ orbifolds, where the integrand decomposes into a sum of orbifold blocks that can be organised into orbits of the Hecke congruence subgroup Γ0(N). As a result, the original modular integral reduces to an integral over the fundamental domain of Γ0(N), which we then evaluate by extending our previous techniques. Our method is applicable, for instance, to the evaluation of one-loop corrections to BPS-saturated couplings in the low energy effective action of closed string models, of quantum corrections to the Kähler metric and, in principle, of the free-energy of superstring vacua.
Highlights
Perturbative one-loop computations in closed string theory typically require integrating over the moduli space of conformal structures on the world-sheet torus
In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus
The Narain lattice does not play any role in the unfolding procedure, T-duality is kept manifest at all steps, a choice of Weyl chamber is not required and the analytic structure of the amplitude is transparent. We generalise this procedure to the case of Abelian ZN orbifolds, where the integrand decomposes into a sum of orbifold blocks that can be organised into orbits of the Hecke congruence subgroup Γ0(N )
Summary
Perturbative one-loop computations in closed string theory typically require integrating over the moduli space of conformal structures on the world-sheet torus. This approach was first discussed in the Physics literature in [14] where the non-universality of gauge threshold corrections for heterotic orbifold compactifications with non-factorisable tori was shown, and was later applied to the computation of quartic gauge couplings [15, 16], to freely-acting orbifolds with partial or total supersymmetry breaking in [17,18,19], as well as in the context of N = 4 topological amplitudes [20] These papers rely on unfolding the fundamental domain FN against the Narain lattice using the traditional implementation of the orbit method, and suffer from the same drawbacks outlined above. Appendix B summarises useful facts on holomorphic modular forms of Hecke congruence subgroups
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