Bubbling saddles of the gravitational index

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×R^4 S 1 × ℝ 4 , solving the supersymmetry equations. We assume a U(1)^3 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) 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 3 , S^2× S^1 S 2 × 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.