Maximal temperature in flux compactifications

Thermal corrections have an important effect on moduli stabilization leading to the existence of a maximal temperature, beyond which the compact dimensions decompactify. In this note, we discuss generality of our earlier analysis and apply it to the case of flux compactifications. The maximal temperature is again found to be controlled by the supersymmetry breaking scale, Tcrit ∼ √ m3/2MP. In string theory gauge and Yukawa couplings are dynamical quantities whose values are determined by expectation values of scalar fields (moduli). In perturbation theory their potential is flat, and their stabilization is a topic of central importance in string theory. Recently, it has been proposed that a complete stabilization of all moduli fields [1] is possible using a combination of fluxes [2] and non-perturbative effects such as D–brane instantons and gaugino condensation [3]. The moduli potentials receive important thermal corrections [4,5]. These corrections destabilize the moduli at sufficiently high temperature, i.e. drive them to the ‘run–away ’ minimum [6] where the compact dimensions decompactify. In order to avoid this cosmological disaster the temperature in the early universe has to be smaller than a maximal temperature Tcrit. In the following we discuss generality of our previous analysis [5] on dilaton destabilization by thermal effects and apply it to the KKLT scenario [1]. We find again that, generically, the maximal temperature is given by the scale of supersymmetry breaking, i.e., Tcrit ∼ √ m3/2MP. Temperature effects in some field theoretical flux compactifications have recently been considered in [7]. Moduli destabilization at high temperature In thermal field theory, the free energy F plays the role of an effective potential. For a Yang-Mills theory at high temperature T, the free energy has a perturbative expansion in terms of the gauge coupling g, F(g,T) = − π2T 4{ α0 + α2g