Superradiance and black resonator strings encounter helical black strings

We construct a cohomogeneity-1 helical black string in six-dimensional Einstein gravity. The helical solution branches from the onset of the gravitational superradiant instability of the equal-spinning Myers-Perry black string. The isometry group of the helical black string is $\mathbb{R}_T \times U(1)_Z \times SU(2)$, where the first two are helical isometries generated by linear combinations of time translation, shifts along the string, and rotation, each of which is individually broken by the superradiant instability. The helical black string is stationary, non-axisymmetric, and has nonzero horizon velocity despite the absence of momentum in the string direction. The entropy of the helical black string is higher than that of the Myers-Perry black string, but lower than cohomogeneity-2 ``black resonator strings'' (recently found) when the solutions overlap in the microcanonical ensemble. The entropy of the helical black string approaches zero when the horizon velocity along the string reaches its maximum given by the speed of light. Nevertheless, we find no evidence for the existence of regular horizonless solutions in this limit.