Moduli stabilization in type II Calabi-Yau compactifications at finite temperature
We consider the type II superstring compactified on Calabi-Yau threefolds at finite temperature. The latter is implemented at the string level by a free action on the Euclidean time circle. We show that all Kahler and complex structure moduli involved in the gauge theories geometrically engineered in the vicinity of singular loci are lifted by the stringy thermal effective potential. The analysis is based on the effective gauged supergravity at low energy, without integrating out the BPS states becoming massless at the singular loci. The universal form of the action in the weak coupling regime at low enough temperature is determined in two cases. Namely the conifold locus, as well as a locus where the internal space develops a genus-g curve of A{N-1} singularities, realizing an SU(N) gauge theory coupled to g hypermultiplets in the adjoint. In general, the favored points of stabilization sit at the intersection of several singular loci. Thus the entire vector multiplet moduli space can be lifted, together with hypermultiplet moduli. The scalars are dynamically stabilized during the cosmological evolution induced by the back-reaction of the thermal effective potential. When the universe expands and the temperature T drops, the scalars converge to minima, with damped oscillations. Moreover, they store an energy density that scales as T^4, which never dominates over radiation. The reason for this is that the mass they acquire at one-loop is of order the temperature scale, which is time-dependent. As an example we analyze the type IIA compactification on a Calabi-Yau space with Hodge numbers h{11}=2 and h{12}=128. In this case, both Kahler moduli are stabilized, where the internal space develops a node and an enhanced SU(2) gauge theory coupled to 2 adjoint hypermultiplets. This shows that in the dual thermal heterotic picture on K3xT^2, the torus modulus and the axio-dilaton are stabilized.