Effective Vilcovisky-De witt action in mutidimensional R2 gravitation

The gauge- and parametrization-invariant Vilcovisky-De Witt effective action (EA) in multidimensional R² gravitation against R₄×T_d–4 background, where R₄ is a four-dimensional flat space and T_d is a d-dimensional torus, is calculated in the one-loop approximation. It is shown that with the help of a limiting procedure the Vilcovisky-De Witt EA in multidimensional Einstein gravitation against the background R₄×T_d–4 can be constructed from the EA found. The question of the stability of spontaneous compactification is analyzed taking into account quantum effects in d=5 R² gravitation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 106–111, October, 1989.I thank I. L. Bukhbinder and I. V. Tyutin for useful discussions.