The gravitational index of 5d black holes and black strings

A bstract The supersymmetric index of 5 d black strings and spinning black holes in M-theory is related to that of 4 d black holes in type IIA supergravity when both theories are compactified on the same Calabi-Yau threefold. We find the finite-temperature saddles for the 5 d gravitational supersymmetric index by uplifting the recently found attractor saddles of the corresponding 4 d index. We study uplifts for two types of geometries: 5 d black holes and 5 d black strings. For 5 d black holes, the uplift guarantees that the index of 4 d and 5 d black holes match. For 5 d black strings, the saddle reproduces the microscopic index at leading order in G N , even without the conventional decoupling limit taken in AdS/CFT. In particular, when the temperature is set to be finite in the 5 d flat space region, the black string index is computed from an asymptotically flat solution where the AdS throat is absent. Further, as the temperature is lowered and eventually becomes infinitesimally small in the flat space region, the solution admits a novel decoupling limit in which the AdS 3 throat takes the form of a finite-temperature BTZ black hole that is known to compute the index in AdS 3 /CFT 2 . This represents the first step towards understanding holography for supersymmetric observables in flat space, away from the decoupling limit.