A spin on Hagedorn temperatures and string stars

We discuss the correspondence between highly excited strings and black holes in the presence of angular momentum. At fixed imaginary angular velocity $\nu$, we show that free strings exhibit a Hagedorn instability due to a thermal-winding mode turning tachyonic. This allows us to determine the exact Hagedorn temperature $\beta_H(\nu)$ for bosonic, type II, and heterotic strings. Using the effective field theory for the thermal-winding mode around the $\nu$-dependent background, we find a novel `rotating string star' saddle (perturbatively in the angular velocity) and study its properties. This configuration describes a self-gravitating bound state of highly excited rotating strings. As in the non-rotating case, the saddle is qualitatively shown to interpolate between the rotating strings phase and a rotating black hole. We also comment on the implications of these results for anti-de Sitter space.