Timelike Supersymmetric Solutions of D=3, N=4 Supergravity

TL;DR

This paper classifies timelike supersymmetric solutions in D=3, N=4 supergravity, showing AdS3 is unique in the gauged case and exploring the structure of solutions in the ungauged theory, including pp-waves.

Contribution

It proves AdS3 is the only timelike supersymmetric solution in the gauged model and characterizes solutions in the ungauged model via holomorphic functions and sigma models.

Findings

AdS3 is the only solution in the gauged supergravity.

Ungauged theory reduces to a sigma model with hyperbolic target space.

Explicit solutions include warped metrics and pp-waves.

Abstract

We study timelike supersymmetric solutions of a $D=3, N=4$ gauged supergravity using Killing spinor bilinears method and prove that AdS$_3$ is the only solution within this class. We then consider the ungauged version of this model. It is found that for this type of solutions, the ungauged theory effectively truncates to a supergravity coupled to a sigma model with a 2-dimensional hyperbolic target space $\mathbb{H}^2$, and all solutions can be expressed in terms of two arbitrary holomorphic functions. The spacetime metric is a warped product of the time direction with a 2-dimensional space, and the warp factor is given in terms of the K\"ahler potential of $\mathbb{H}^2$. We show that when the holomorphic function that determines the sigma model scalar fields is not constant, the metric on the sigma model target manifold becomes part of the spacetime metric. We then look at some special choices for these holomorphic functions for which the spacetime metric and the Killing spinors are only radial dependent. We also derive supersymmetric null solutions of the ungauged model which are pp-waves on the Minkowski spacetime.