Timelike supersymmetric solutions of $ D=3$, $ N=4$ supergravity

We study timelike supersymmetric solutions of a $ D=3$, $ N=4$ gauged supergravity using the Killing spinor bilinears method and prove that $ {\mathrm{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 solution, the ungauged theory effectively truncates to a supergravity coupled to a sigma model with a two-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 two-dimensional space, and the warp factor is given in terms of the Kähler 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.