AdS$_5$ black strings in the stu model of FI-gauged $N=2$ supergravity

We analytically construct asymptotically AdS$_5$ black string solutions starting from the four-dimensional domain wall black hole of arXiv:0911.4926. It is shown that its uplift gives a black string in $d=5$ minimal gauged supergravity, with momentum along the string. Applying instead the residual symmetries of $N=2$, $d=4$ Fayet-Iliopoulos-gauged supergravity discovered in arXiv:1606.05160 to the domain wall seed leads, after uplifting, to a dyonic black string that interpolates between AdS$_5$ and $\text{AdS}_3\times\text{H}^2$ at the horizon. A Kaluza-Klein reduction of the latter along an angular Killing direction $\phi$ followed by a duality transformation yields, after going back to five dimensions, a black string with both momentum along the string and rotation along $\phi$. This is the first instance of using solution-generating techniques in gauged supergravity to add rotation to a given seed. These solutions all have constant scalar fields. As was shown in hep-th/0302218, the construction of supersymmetric static magnetic black strings in the FI-gauged stu model amounts to solving the $\text{SO}(2,1)$ spinning top equations, which descend from an inhomogeneous version of the Nahm equations. We are able to solve these in a particular case, which leads to a generalization of the Maldacena-Nu\~nez solution.