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

TL;DR

The paper expands the landscape of gravitational index saddles in five-dimensional supergravity by systematically constructing Euclidean multicenter solutions—including black holes, rings, lenses, and Saturns—in both flat and AdS$_3\times S^2$ backgrounds. It details a robust 4d/5d uplift mechanism that desingularizes certain 4d configurations, enabling smooth 5d saddles and revealing an index enigma where less symmetric saddles can dominate the index compared to more symmetric counterparts. The authors derive explicit smoothness and integrability conditions, compute the on-shell actions as entropies of the corresponding extremal objects, and analyze the associated moduli spaces, including their temperature dependence and wall-crossing behavior. They show, in flat space and in AdS, that multiple distinct geometries with different horizon topologies can contribute to the same index, with dominance depending on charges and ensemble, thereby refining the dictionary between gravitational saddles and microscopic index computations. The work suggests an algorithmic path to discover further 5d saddles beyond uplifted multicenter constructions and points toward broader implications for protected indices in higher-dimensional gravity and their CFT duals.

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.

Figures (2)

• Figure 1: Examples of allowable values of $x_{12}$ distance as a function of $\beta$ for the case of black Saturn in asymptotically flat $5d$ space, as determined by the smoothness of the saddle, for two different choices of total charges. In both figures, the dashed lines represent the distance which matches the extremal bound state distance $x_{12}^{*} = -\langle \Gamma_1 , \Gamma_2 \rangle/\langle \Gamma_1, h \rangle$. Left: For the choice of charges $\Gamma_1 = (1,9,3,3)$, $\Gamma_2 = (0,6,3,3)$, as we lower the temperature, the space of solutions shrinks until it collapses on the extremal value of the distance. Right: For the choice of charges $\Gamma_1 = (1,2,1,20)$, $\Gamma_2 = (0,13,10,5)$, as we lower the temperature, the moduli space becomes disconnected, leading to two possible extremal solutions. The solution corresponding to the left peak is a scaling solution where all base-space distances shrink as $\sim 1/\beta$.
• Figure 2: Example of allowable values of $x_{12}$ distance as a function of $\beta$ for the case of two BMPV black holes in AdS$_3 \times S^2$ as determined by smoothness of the saddle. The dashed lines represent the distance which matches the extremal bound state distance $x_{12}^{*} = -\langle \Gamma_1 , \Gamma_2 \rangle/\langle \Gamma_1, h \rangle$. For the charges $\Gamma_1 = (-2,12,3,3)$, $\Gamma_2 = (2,6,3,3)$, as we lower the temperature, the space of solutions "shrinks" until it collapses on the extremal value of the distance. For high temperatures, we pass the critical temperature after which the saddle cannot be smooth, and, presumably, we lose its contribution to the path integral.