The gravitational index of a small black ring

Certain supersymmetric elementary string states with angular momentum can be viewed as small black rings in a five-dimensional string theory. These black rings have zero area event horizon. The 4D-5D connection relates these small rings to small black holes without angular momentum in one less dimension. Recent works have proposed saddle solutions that compute the supersymmetric index for small black holes using gravitational path integral. In this paper, we propose an analogous saddle solution for a five-dimensional small black ring. The dominant contribution comes from a black ring saddle that rotates in both independent planes in five dimensions and has a finite area event horizon. We also write the saddle solution as a three-center Bena-Warner solution.