Novel black saddles for 5d gravitational indices and the index enigma

TL;DR

This paper constructs new supersymmetric black hole solutions in five-dimensional supergravity that serve as saddle points for gravitational indices, revealing a novel 'index enigma' where less symmetric solutions dominate.

Contribution

It introduces a series of novel Euclidean multi-black-hole solutions in 5d supergravity that contribute to gravitational indices and uncovers the 'index enigma' phenomenon.

Findings

Constructed supersymmetric black hole, black ring, black Saturn, and black lens solutions.

Found that less symmetric black ring saddles dominate in flat space index calculations.

Demonstrated on-shell actions match black hole entropies and are temperature-independent.

Abstract

We construct a series of novel Euclidean multi-black-hole, black ring, black Saturn, and black lens solutions to $5d$ supergravity that contribute as saddle-points to the $5d$ gravitational supersymmetric index, either in asymptotically flat space or in asymptotically AdS$_3\times S^2$. All these solutions are supersymmetric, have finite temperature, and an appropriate angular velocity turned on that makes fermionic fields periodic around the thermal circle. They contribute either to the helicity supertrace of supergravity in $5d$ flat space or to the elliptic genus of a supergravity theory in AdS$_3 \times S^2$. Their on-shell actions are independent of temperature, as consistent with the computation of a protected index, and equal to the entropy of the corresponding extremal black object. Our construction relies on uplifting saddles that can be singular in $4d$, but which are desingularized in $5d$. The resulting saddles exhibit a novel ``index enigma'', not encountered in previous Lorentzian solutions. One example of this enigma is that, in the computation of the index in asymptotically flat space, less symmetric black ring saddles dominate over the contributions from $5d$ black holes.