ar X iv : h ep - t h / 97 05 03 6 v 1 6 M ay 1 99 7 Sigma - model for the Generalized Composite p - branes

Multidimensional gravitational model containing several dilatonic scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M 0 × M 1 ×. .. × M n , where M i are Einstein spaces (i ≥ 1). When metric is block-diagonal, all fields and scale factors of the metric depend (essentially) on the point of M 0 and for the forms composite (electromagnetic) p-brane Ansatz is adopted the model is reduced to the gravitating self-interacting sigma-model with certain constraints imposed. In pure electric and magnetic cases the number of these constraints is n 1 (n 1 −1)/2 where n 1 is number of 1-dimensional manifolds among M i. In the " electromagnetic " case for dimM 0 = 1, 3 additional n 1 constraints appear. A family of " Majumdar-Papapetrou type " solutions governed by a set of harmonic functions is obtained, when all factor-spaces M ν are Ricci-flat. These solutions are generalized to the case of non-Ricci-flat M 0 when also some additional " internal " Einstein spaces of non-zero curvature are added to M. As an example exact solutions for D = 11 supergravity and related 12-dimensional theory are presented.