Effective Action in Multidimensional (Super)Gravities and Spontaneous Compactification (Quantum Aspects of Kaluza-Klein Theories)

The review of modern status of problem of quantum effects in Kaluza-Klein theories is given. The effective action (EA) in multidimensional (super)gravities (SG's) on the compactified background is investigated. The standard gauge dependent EA in d=5 Einstein gravity and d=5 R²-gravity on the background R₄ × S₁, where R₄ is 4-dimensional space, S₁ is one-dimensional sphere is calculated. Gauge and parametrization independent Vilcovisky-De Witt EA in d=5 Einstein gravity and d=5 R²-gravity on the background R₄×S₁ at zero and non-zero temperature is obtained. We have found that there are no physically acceptable self-consistent solutions of the form R₄×S₁ at the one-loop level in d=5 Einstein gravity. We calculated also EA for arbitrary multidimensional SG on the background R₄×T_d-n where T_d is d-dimensional torus as expansion on the curvature and its derivatives. The mechanizm of induced of four-dimensional gravity with zero Λ-term is proposed. The Vilcovisky-De Witt EA in d=5 SG's on the background R₄×S₁ at non-zero temperature is obtained. The three gauge parameter dependent off-shell EA in N=2, d=5 gauged SG on R₄⁰×S₁ where R₄⁰ is flat four-dimensional space is calculated. The expression for vacuum energy for bosonic string with torus compactification is presented. Vacuum energy for superstrings with supersymmetry broken as the result of choice of boundary conditions on background R₄×T₆ is calculated.