Two-Loop Approach to the Effective Action in Quantum Gravity

For the first time we present the general formalism and results of calculation of the two-loop effective action in Einstein quantum gravity on the background M N × T k , where M N is Minkowski space and T k is a k-dimensional torus. We discuss the case of a zero cosmological constant as well as of a nonzero one. The method of calculating variations of the action on a metric tensor and the technique of calculating momentum integrals in dimensional regularization are presented. Some applications to spontaneous compactification are discussed, as well as some prospects.