Sigma-model approach in general relativity and its generalizations

Abstract

Construction of exact solutions of Einstein equations in General Relativity and its generalizations, such as supergravities and string inspiredmodels in various dimensions is an important problemof theoretical andmathematical physics. Herewe reviewone of themost efficient tools which is applicable to large class of theories involvingmetric, scalar, vector and antisymmetric form fields without higher-derivative terms, scalar potentials and cosmological constant. This includes pure gravity in arbitrary dimensions, bosonic parts of supergravities, as well as many non-supersymmetric gravity-matter systems. The method is based on dimensional reduction of D-dimensional Einstein equations assuming the solutions to depend only on three independent variables in away that the field equations reduce to those of the three-dimensional gravity coupled scalar sigma-model. Typically this amounts to starting with the Einstein-Hilbert action coupled to scalar-vector/antisymmetric formsector withMaxwell-dilaton type action and restricting to field configurations admitting D−3 commuting Killing vectors. The corresponding dimensional reduction is commonly referred as toroidal.