Symplectic Gravity Models in Four, Three and Two Dimensions

TL;DR

This paper develops a comprehensive framework for symplectic gravity models in various dimensions, revealing their symmetries, sigma-model representations, and solution-generating techniques, advancing the understanding of coupled vector and scalar fields in gravity theories.

Contribution

It introduces a sigma-model and Kähler formalism for symplectic gravity models, including the construction of chiral matrices and transformations, extending the mathematical tools for analyzing these theories.

Findings

The equations of motion exhibit Sp(2n, R) symmetry in four dimensions.

A sigma-model representation with target space invariant under Sp[2(n+1), R] is established.

Explicit chiral matrices and transformations like Kramer-Neugebauer are constructed.

Abstract

A class of the $D=4$ gravity models describing a coupled system of $n$ Abelian vector fields and the symmetric $n \times n$ matrix generalizations of the dilaton and Kalb-Ramond fields is considered. It is shown that the Pecci-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 $\sigma$-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 K\"ahler formalism based on the use of the Ernst $(n+1) \times (n+1)$ 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.