Square of General Relativity

Abstract

We consider dilaton--axion gravity interacting with $p\;\, U(1)$ vectors ($p=6$ corresponding to $N=4$ supergravity) in four--dimensional spacetime admitting a non--null Killing vector field. It is argued that this theory exibits features of a ``square'' of vacuum General Relativity. In the three--dimensional formulation it is equivalent to a gravity coupled $\sigma$--model with the $(4+2p)$--dimensional target space $SO(2,2+p)/(SO(2)\times SO(2+p))$. K\"ahler coordinates are introduced on the target manifold generalising Ernst potentials of General Relativity. The corresponding K\"ahler potential is found to be equal to the logarithm of the product of the four--dimensional metric component $g_{00}$ in the Einstein frame and the dilaton factor, independently on presence of vector fields. The K\"ahler potential is invariant under exchange of the Ernst potential and the complex axidilaton field, while it undergoes holomorphic/antiholomorphic transformations under general target space isometries. The ``square'' property is also manifest in the two--dimensional reduction of the theory as a matrix generalization of the Kramer--Neugebauer map.