Symmetries of the stationary Einstein--Maxwell--dilaton theory

TL;DR

This paper analyzes the symmetry structures of the stationary Einstein--Maxwell--dilaton system, revealing how the symmetry algebra varies with the dilaton coupling and enabling generation of new solutions through discrete maps.

Contribution

It identifies the symmetry algebra for general dilaton coupling and shows its enhancement at specific critical values, connecting these to known theories and providing methods to generate new solutions.

Findings

Symmetry algebra is isomorphic to a maximal solvable subalgebra of sl(3,R) for general coupling.

At specific couplings, the algebra enlarges to full sl(3,R) and su(2,1)×R.

Discrete maps are found to generate new solutions in dilaton gravity.

Abstract

Gravity coupled three--dimensional $\sigma$--model describing the stationary Einstein--Maxwell--dilaton system with general dilaton coupling is studied. Killing equations for the corresponding five--dimensional target space are integrated. It is shown that for general coupling constant $\alpha$ the symmetry algebra is isomorphic to the maximal solvable subalgebra of $sl(3,R)$. For two critical values $\alpha =0$ and $\alpha =\sqrt{3}$, Killing algebra enlarges to the full $sl(3,R)$ and $su(2,1)\times R$ algebras respectively, which correspond to five--dimensional Kaluza--Klein and four--dimensional Brans--Dicke--Maxwell theories. These two models are analyzed in terms of the unique real variables. Relation to the description in terms of complex Ernst potentials is discussed. Non--trivial discrete maps between different subspaces of the target space are found and used to generate new arbitrary--$\alpha$ solutions to dilaton gravity.