The gravitational index of 5d black holes and black strings

TL;DR

This paper explores the relationship between 5d black hole and black string indices in M-theory, establishing connections with 4d black hole indices and analyzing finite-temperature saddles, advancing understanding of holography in flat space.

Contribution

It introduces a method to uplift 4d attractor saddles to 5d, linking black hole and black string indices, and investigates finite-temperature solutions beyond the conventional decoupling limit.

Findings

Matching indices for 4d and 5d black holes via uplift.

Reproducing the microscopic index for black strings at leading order.

Identifying a novel decoupling limit with a finite-temperature BTZ black hole.

Abstract

The supersymmetric index of 5d black strings and spinning black holes in M-theory is related to that of 4d 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 5d gravitational supersymmetric index by uplifting the recently found attractor saddles of the corresponding 4d index. We study uplifts for two types of geometries: 5d black holes and 5d black strings. For 5d black holes, the uplift guarantees that the index of 4d and 5d black holes match. For 5d 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 5d 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.