Black holes from the gravitational path integral: supersymmetric indices and precision holography

TL;DR

This work develops a comprehensive program to connect microscopic counts of supersymmetric black hole microstates with macroscopic gravitational descriptions using the gravitational path integral. It advances the gravitational index program by formulating and evaluating complex Euclidean saddles in five-dimensional gauged supergravity, including higher-derivative corrections, and matches these semiclassical results to the Cardy-like limits of dual four-dimensional superconformal indices. It further builds a robust holographic dictionary for anomaly-driven, four-derivative corrections, derives corrected black hole entropies via constrained Legendre transforms and near-horizon Wald methods, and extends the analysis to asymptotically flat cases with a rich set of bubbling saddles and localization techniques in odd dimensions. The results provide precision holography tests at subleading orders in large-N and offer a versatile framework to study both AdS5 black holes and flat-space gravitational saddles, with implications for quantum gravity, holography, and black hole microphysics. Overall, the thesis demonstrates that higher-derivative corrections and novel gravitational saddles reproduce refined field theory predictions for black hole entropy and indices, reinforcing the AdS/CFT correspondence and elucidating quantum gravity corrections in controlled settings.

Abstract

The counting of microstates of supersymmetric black holes with anti-de Sitter or flat asymptotics is obtained by computing a supersymmetric index in a weakly coupled string theory or a dual superconformal field theory. These indices are protected observables, whose value can be reliably extrapolated from weak to strong coupling, where the gravitational description applies. In this Thesis, after a broad introductory review, we discuss recent progress in formulating such protected observables directly within the gravitational theory, via the Euclidean path integral. In the semiclassical limit the index reduces to a sum over complex Euclidean saddles. These saddles are supersymmetric but ''non-extremal'', and arise in both anti-de Sitter and flat spaces. In the holographic setting, we investigate four-derivative corrections to the thermodynamics of AdS$_5$ black holes. Using off-shell methods, we construct the corrected action of five-dimensional gauged supergravity. We then evaluate the corrected on-shell action of supersymmetric AdS$_5$ black holes and find exact agreement with a Cardy-like limit of the superconformal index of the dual conformal field theory. By a Legendre transform of the action, we obtain the corrected black hole entropy, and we confirm this result by applying Wald's formula to the corrected near-horizon geometry. We then turn to the gravitational index with asymptotically flat boundary conditions. We uncover a broad family of novel saddles and present a general classification based on their rod structure, which characterizes their topology. These solutions may feature multiple horizons or three-dimensional bubbles with lens space topology, and allowing for conical singularities yields further geometries, involving spindles and branched spheres. For the simpler geometries, the on-shell action is computed using an odd-dimensional version of equivariant localization.

Figures (8)

• Figure 1: Closed strings (in green) form closed loops, while open strings (blue) are forced to end on D-branes.
• Figure 2: The two pictures of a black hole in string theory: the horizontal axis represents the effective coupling $g_{\rm s} N$. When $g_{\rm s} N \ll 1$ (left) the effective description is that of fluctuating strings and branes, carrying the quantum number $N$, in a weakly coupled string theory; while for $g_{\rm s} N \gg 1$ (right) the interactions become strong and a black hole with charge $N$ is formed. The string theory is strongly coupled, but the supergravity description is reliable.
• Figure 3: AdS/CFT is a gauge/gravity duality. After a suitable decoupling limit, a stack of $N$ D$3$-branes admits two equivalent descriptions: Type IIB string theory on AdS$_5$$\times$$S^5$ and ${\cal N}=4$, $D=4$ SYM. When $g_{\rm s} N \ll 1$ (left) the SYM sitting on the brane worldvolume is weakly coupled. In the opposite regime, $g_{\rm s} N \gg 1$ (right), the SYM is strongly coupled, but the supergravity approximation is reliable.
• Figure 4: The quivers describing the low-energy limit of D3-branes probing a $\mathbb{C}^3/\mathbb{Z}_\nu$ singularity. Left: a generic ${\rm SU}(N)$ node $\alpha$ is connected to the node $(\alpha+1)$ by a doublet of outgoing arrows and to the node $(\alpha+2)$ by an incoming arrow (an arrow pointing from node $\alpha$ to node $\beta$ denotes a chiral superfield in the $({\bf N}, {\bf \bar{N}})$ bifundamental representation of ${\rm SU}(N)_{\alpha}\times {\rm SU}(N)_{\beta}$). Right: the case $\nu = 5$.
• Figure 5: 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 $z_s$.