Symplectic gravity models in four, three and two dimensions

Abstract

A class of the D=4 gravity models describing a coupled system of n Abelian vector fields and the symmetric n×n matrix generalizations of the dilaton and Kalb–Ramond fields is considered. It is shown that the Peccei–Quinn axion matrix can be entered and the resulting equations of motion possess the Sp(2n,R) symmetry in four dimensions. The stationary case is studied. It is established that the theory allows a σ-model representation with a target space which is invariant under the Sp[2(n+1),R] group of isometry transformations. The chiral matrix of the coset Sp[2(n+1),R]/U(n+1) is constructed. A Kähler formalism based on the use of the Ernst (n+1)×(n+1) complex symmetric matrix is developed. The stationary axisymmetric case is considered. The Belinsky–Zakharov chiral matrix depending on the original field variables is obtained. The Kramer–Neugebauer transformation, which algebraically maps the original variables into the target space ones, is presented.