We study a system of gravity and free massless scalar fields minimally coupled to gravity in a 7-dimensional background which is a direct product of a 4-dimensional Minkowski space and a 3-dimensional homogeneously deformed three-sphere. Compactification is caused by the vacuum energy of scalar fields. The effective potential as a function of two parameters (scale and deformation) is calculated numerically after dimensional regularization. We find the effective potential decreases rapidly toward negative infinity in both prolate and oblate directions. The classical curvature, however, can balance the quantum effect and yields three extrema. In addition to the round $S^3$ solution in the quantum-corrected field equations, we find two deformed ones. One of the deformed solutions corresponds to the local minimum of the total potential. The round three-sphere solution, however, corresponds to the local maximum of that. More scalar fields can enlarge the scale of the internal space but not affect the shape. This serves as an example of gauge symmetry breaking by deformation of the internal space in multidimensional theories. The stability of these background solutions is discussed but not established conclusively. A discussion of four different analytic-continuation procedures is presented in one of the appendixes.

• Received 6 October 1986

DOI:https://doi.org/10.1103/PhysRevD.36.397