Calculation of gauge couplings and compact circumferences from self-consistent dimensional reduction

Abstract

We consider a system of gravity plus free massless matter fields in 4 + N dimensions, and look for solutions in which N dimensions form a compact curved manifold, with the energy-momentum tensor responsible for the curvature produced by quantum fluctuations in the matter fields. For manifolds of sufficient symmetry (including spheres, CPN, and manifolds of simple Lie groups) the metric depends on only a single multiplicative parameter ϱ2, and the field equations reduce to an algebraic equation for ϱ, involving the potential of the matter fields in the metric of the manifold. With a large number of species of matter fields, the manifold will be larger than the Planck length, and the potential can be calculated using just one-loop graphs. In odd dimensions these are finite, and give a potential of form CN/ϱ4. Also there are induced Yang-Mills and Einstein-Hilbert terms in the effective 4-dimensional action, proportional to additional numerical coefficients, DN and EN. General formulas are given for the gauge coupling g2 in terms of CN and DN, and the ratio ϱ2/8πG in terms of CN and EN. Numerical values for CN, DN, and EN are obtained for scalar and spinor fields on spheres of odd dimensionality N. It is found that the potential, g2 and ϱ2/8πG can all be positive but only when the compact manifold has N = 3 + 4 k dimensions. (The positivity of the potential is needed for stability of the sphere against uniform dilations or contractions). In this case, solutions exist either for spinor fields alone or for suitable mixes of spinor and scalar fields provided the ratio of the number of scalar fields to the number of fermion fields is not too large. Numerical values of the O(N + 1) gauge couplings and 8φG/ϱ2 are calculated for illustrative values of the numbers of spinor fields. It turns out that large numbers of matter fields are needed to make these parameters reasonably small.