A generalised Garfinkle–Vachaspati transform

Abstract

The Garfinkle-Vachaspati transform is a deformation of a metric in terms of a null, hypersurface orthogonal, Killing vector $k^\mu$. We explore a generalisation of this deformation in type IIB supergravity taking motivation from certain studies of the D1-D5 system. We consider solutions of minimal six-dimensional supergravity admitting null Killing vector $k^\mu$ trivially lifted to type IIB supergravity by the addition of four-torus directions. The torus directions provide covariantly constant spacelike vectors $l^\mu$. We show that the original solution can be deformed as $g_{\mu \nu} \to g_{\mu \nu} + 2 \Phi k_{(\mu}l_{\nu)}, C_{\mu \nu} \to C_{\mu \nu} - 2 \Phi k_{[\mu}l_{\nu]}$, provided the two-form supporting the original spacetime satisfies $i_k (dC) = - d k$, and where $\Phi$ satisfies the equation of a minimal massless scalar field on the original spacetime. We show that the condition $i_k (dC) = - d k$ is satisfied by all supersymmetric solutions admitting null Killing vector. Hence all supersymmetric solutions of minimal six-dimensional supergravity can be deformed via this method. As an example of our approach, we work out the deformation on a class of D1-D5-P geometries with orbifolds. We show that the deformed spacetimes are smooth and identify their CFT description. Using Bena-Warner formalism, we also express the deformed solutions in other duality frames.