Kerr-NUT-AdS metrics and string theory

Abstract

With the advent of supergravity and superstring theory, it is of great importance to study higher-dimensional solutions to the Einstein equations. In this dissertation, we study the higher dimensional Kerr-AdS metrics, and show how they admit further generalisations in which additional NUT-type parameters are introduced. The choice of coordinates in four dimensions that leads to the natural inclusion of a NUT parameter in the Kerr-AdS solution is rather well known. An important feature of this coordinate system is that the radial variable and the latitude variable are placed on a very symmetrical footing. The NUT generalisations of the highdimensional Kerr-AdS metrics obtained in this dissertation work in a very similar way. We first consider the Kerr-AdS metrics specialised to cohomogeneity 2 by appropriate restrictions on their rotation parameters. A latitude coordinate is introduced in such a way that it, and the radial variable, appeared in a very symmetrical way. The inclusion of a NUT charge is a natural result of this parametrisation. This procedure is then applied to the general D dimensional Kerr-AdS metrics with cohomogeneity [D/2]. The metrics depend on the radial coordinate r and [D/2] latitude variables µi that are subject to the constraint Ei µ2i = 1. We find a coordinate reparameterisation in which the µi variables are replaced by [D/2]−1 unconstrained coordinates yα, and put the coordinates r and yα on a parallel footing in the metrics, leading to an immediate introduction of ([D/2]−1) NUT parameters. This gives the most general Kerr-NUT-AdS metrics in D dimensions. We discuss some remarkable properties of the new Kerr-NUT-AdS metrics. We show that the Hamilton-Jacobi and Klein-Gordon equations are separable in Kerr- NUT-AdS metrics with cohomogeneity 2. We also demonstrate that the general cohomogeneity-n Kerr-NUT-AdS metrics can be written in multi-Kerr-Schild form. Lastly, We study the BPS limits of the Kerr-NUT-AdS metrics. After Euclideanisation, we obtain new families of Einstein-Sassaki metrics in odd dimensions and Ricci-flat metrics in even dimensions. We also discuss their applications in String theory.