Bubbling saddles of the gravitational index

TL;DR

The paper analyzes the five-dimensional supergravity path integral for a grand-canonical supersymmetric index, revealing a rich landscape of semiclassical saddles with bubbling topology organized by a rod-structure under a ${\rm U}(1)^3$ symmetry. By formulating complex finite-temperature solutions and classifying fixed loci into horizon and bubbling rods, it derives on-shell actions and thermodynamic relations that reproduce known Lorentzian BPS black rings and black lenses in extremal limits, while also describing horizonless bubbling solitons in a different limit. The key contributions include a detailed topological taxonomy of bolts and rods, explicit expressions for the on-shell action in multi-center settings (notably three-center), and a clear bridge between Euclidean saddles and Lorentzian microstate geometries. This framework advances the understanding of the gravitational index, microstate counting, and phases of supersymmetric gravitating systems, and lays groundwork for future localization approaches and microscopic comparisons.

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.

Figures (2)

• Figure 1: Generic representation of the rod structure. The compact rod $I_a$ joins the Gibbons-Hawking centers placed at $z_a$ and $z_{a+1}$, while the dots represents any possible arrangement of compact rods. The first and last ones are semi-infinite segments originating from $z_1$ and $z_s$.
• Figure 3: Rod structure of the three-center solution, with the bubbling rod $I_1$ and the horizon rod $\mathcal{H}$. For each rod we indicate the corresponding rod vector.