Heterotic Moduli Stabilisation

We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactified on manifolds which are Calabi-Yau up to alpha' effects. We review how to fix all geometric and bundle moduli in a supersymmetric way by fractional fluxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to W, non-perturbative and threshold effects. We then show that alpha' corrections to K lead to new stable Minkowski (or dS) vacua where the complex structure moduli Z and the dilaton are fixed supersymmetrically, while the fixing of the Kahler moduli at a lower scale leads to spontaneous SUSY breaking. The minimum lies at moderately large volumes of all geometric moduli, at a perturbative string coupling and at the right value of the GUT coupling. We also give a dynamical derivation of anisotropic compactifications which allow for gauge coupling unification around 10^16 GeV. The gravitino mass can be anywhere between the GUT and TeV scale depending on the fixing of the Z-moduli. In general, these are fixed by turning on background fluxes, leading to a gravitino mass around the GUT scale since the heterotic 3-form flux does not contain enough freedom to tune W to small values. Moreover accommodating the observed value of the cosmological constant (CC) is a challenge. Low-energy SUSY could instead be obtained in particular situations where the gauge bundle is holomorphic only at a point-like sub-locus of Z-moduli space, or where the number of Z-moduli is small (like orbifold models), since in these cases one may fix all moduli without turning on any quantised flux. However tuning the CC is even more of a challenge in these cases. Another option is to focus on non-complex manifolds since these allow for new geometric fluxes which can be used to tune W and the CC, even if their moduli space is presently only poorly understood.