Rotating AdS$_3 \times$S$^3$ and Dyonic Strings from 3-Dimensions

We make a general Killing spinor analysis of a particular $D=3, \, N=4$ gauged supergravity that comes from a consistent S$^3$ reduction of $D=6, \, N=(1,0)$ supergravity coupled to a single chiral tensor multiplet. We then focus on its supersymmetric solutions with a null Killing vector and find three new ones. Two of these, namely the null warped AdS$_3$ (also known as the Schr\"odinger spacetime) and the charged domain wall solutions, admit non-trivial gauge fields which give rise to rotating solutions in 6-dimensions. The uplift of the first one produces an interesting AdS$_3 \times$S$^3$ background with a non-trivial rotation in the $U(1)$ fiber direction of the S$^3$ which retains the Schr\"odinger scale invariance that the seed solution has. The second one leads to the well-known rotating dyonic string solution. Finally, the uplift of the third one, which is a domain wall solution with no gauge fields, results in a distribution of dyonic strings.