Bounds on complex structure moduli values for perturbative control

Abstract

String compactification in the framework of the low energy effective supergravity requires the perturbative control in both the large volume and the weak coupling expansions. However, when the complex structure moduli couple to some lattice structure, the Sp$(2(h^{2,1}+1))$ symmetry of the tree level Kähler potential allows the correction to the Kähler potential to diverge in the large field limit of the complex structure moduli, resulting in the breakdown of the perturbative control. Here the lattice structure naturally appears in the presence of a tower of states like the Kaluza-Klein or the string modes, an essential ingredient of the distance conjecture. The similar situation can be found from the axio-dilaton contribution to the corrected Kähler potential, where the SL$(2,\mathbb{Z})$ symmetry as well as the coupling between the axio-dilaton and the lattice structure allow the correction to diverge in the weak coupling limit. In order to keep the perturbative control, the values of the complex structure moduli as well as the dilaton must have the upper bound, which is determined by the volume of the internal manifold and the string coupling constant, hence the Kaluza-Klein and the string mass scales. The form of the bounds are quite similar to that given by the distance conjecture, both prevents the descent of a tower of states.

24 pages, version published in CQG

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