Bubbling saddles of the gravitational index

TL;DR

This paper uncovers complex semiclassical saddle solutions with bubbling topology in five-dimensional supergravity, analyzing their contributions to the gravitational index and connecting to known black ring and horizonless solutions.

Contribution

It introduces a general construction of bubbling saddle solutions with diverse topologies in five-dimensional supergravity and evaluates their impact on the gravitational index.

Findings

Identified new semiclassical saddles with bubbling topology.

Connected bubbling solutions to known black ring and black lens solutions.

Analyzed limits leading to Lorentzian black hole and horizonless solutions.

Abstract

We consider the five-dimensional supergravity path integral that computes a supersymmetric index, and uncover a wealth of semiclassical saddles with bubbling topology. These are complex finite-temperature configurations asymptotic to $S^1\times\mathbb{R}^4$, solving the supersymmetry equations. We assume a ${\rm U}(1)^3$ symmetry given by the thermal isometry and two rotations, and present a general construction based on a rod structure specifying the fixed loci of the ${\rm U}(1)$ isometries and their three-dimensional topology. These fixed loci may correspond to multiple horizons or three-dimensional bubbles, and they may have $S^3$, $S^2\times S^1$, or lens space topology. Allowing for conical singularities gives additional topologies involving spindles and branched spheres or branched lens spaces. As a particularly significant example, we analyze in detail the configurations with a horizon and a bubble just outside of it. We determine the possible saddle-point contribution of these configurations to the gravitational index by evaluating their on-shell action and the relevant thermodynamic relations. We also spell out two limits leading to well-defined Lorentzian solutions. The first is the extremal limit, which gives the known BPS black ring and black lens solutions. The on-shell action and chemical potentials remain well-defined in this limit and should thus provide the contribution of the black ring and black lens to the gravitational index. The second is a limit leading to horizonless bubbling solutions, which have purely imaginary action.