Gravitational Casimir energy in non-Abelian Kaluza-Klein theories

The Casimir energy of the gravitational field in certain Kaluza-Klein theories is computed. Specifically, the one-loop quantum effective potential of the gravitational field on the background manifold (Minkowski space)×(N-sphere) is evaluated as a function of the radius of the N-sphere. (N is restricted to be odd for technical reasons.) Minima of the (real part of the) potential for which the observed cosmological constant is zero are found numerically for N=13, 15, 17, 19, and 21. The potential has an imaginary part which is attributed to ‘‘tachyonic’’ terms in the mode sum, although it is argued that the Faddeev-Popov ghost modes do not contribute to this imaginary part.