The strong-CP problem and target-space modular invariance in 4D strings

We consider the problem of strong-CP violation in the context of four-dimensional strings. We remark that the action of target-space modular invariance on the massless fermions (a chiral U(1) Kaehler rotation) plays the role of a Peccei-Quinn symmetry in a large class of four-dimensional strings. The would-be axions are provided by massless scalar fields with vanishing potential and couplings to massive coloured particles. An invisible axion mechanism of the type discussed by Kim and Shifman, Vainshtein and Zakharov is at work, guaranteeing the smallness of the QCD Θ-parameter. In addition, by studying a class of supersymmetry (SUSY)-breaking effective actions invariant under target-space modular invariance, we show that the dynamical minima are such that the infinite tower of massive string excitations does not give any contribution to the strong Θ-parameter. Moreover these vacua lead to real SUSY-breaking soft terms. This would provide an explanation for the smallness of the electric dipole moment of the neutron which tends to be two to three orders of magnitude too large in generic supersymmetric models. (orig.)