Spontaneous compactification in generalized Candelas-Weinberg models

In their recent work on the dimensional reduction, Candelas and Weinberg considered a model which is compactified into a direct product space of the Minkowski space (M₄) and an N-dimensional sphere (S_N). In the present paper we investigate generalized models of their type which are compactified into M₄ × S_M × S_N and M₄ × S_M × CP₂. The compactification is caused by the quantum loop effect due to a large number of matter fields. The conditions for the vacuum stability are studied. Numerical computation of the loop effect is undertaken, and it is shown that some of the models of the type M₄ × S_M × S_N admit a stable solution which has finite circumferences of both of the extra spaces and positive coupling constants of the Einstein-Yang-Mills theory in four dimensions.