Bubbling saddles of the gravitational index

Davide Cassani; Alejandro Ruipérez; Enrico Turetta

doi:10.21468/SciPostPhys.19.5.134

paper

Bubbling saddles of the gravitational index

arXiv:2507.12650 · doi:10.21468/SciPostPhys.19.5.134

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 $S^{1} \times R^{4}$ , solving the supersymmetry equations. We assume a $U (1)^{3}$ 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 $U (1)$ 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.

arXiv comment69 pages, 3 figures; matches published version

Bubbling saddles of the gravitational index

Abstract

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