Cosmological constant, dilaton field and Freund–Rubin compactification

We discuss Freund–Rubin compactification with a cosmological constant and the dilaton field, and examine the stability of spacetime at low energy. Minkowski or de Sitter spacetime can be obtained if the dilation field is turned off, but only anti-de Sitter spacetime is realized in the presence of the dilaton field. The stability of spacetime depends on the dimensionality of spacetime and the compactified space, and the coupling constant of the dilaton field a. In the a=1 case, which corresponds to superstring theories, the anti-de Sitter vacuum is stable at least in the linear level.