Abstract
We obtain an expression for the one-loop effective potential coming from quantum fluctuations of massless scalars and spin-(1/2 fermions in a higher-dimensional manifold of product form ${M}^{4}$\ifmmode\times\else\texttimes\fi{}${S}^{M}$\ifmmode\times\else\texttimes\fi{}${S}^{N}$. In contrast with previous calculations, we consider the case in which the total number of dimensions is even, since this is the relevant case for superstring compactification. A detailed calculation for a ten-dimensional spacetime with two internal three-spheres is performed, and an approximate static solution for the geometry is found, with the two internal radii being of the order of the Planck length for a sufficiently large number of matter fields. We study the stability of this solution and make some remarks on the possible cosmological implications of our results.
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