Euclidean Black Saddles and AdS$_4$ Black Holes

TL;DR

The paper constructs and analyzes Euclidean, locally AdS$_4$ supersymmetric solutions (black saddles) in the STU model, whose boundary is $S^1\times\Sigma_{\mathfrak{g}}$ and which cap off smoothly in the interior. It derives a detailed UV/IR holographic dictionary, performs holographic renormalization (including a Legendre transform for alternate quantization), and proves a precise match between the on-shell action of these saddles and the topologically twisted ABJM index in the planar limit for general magnetic fluxes and flavor deformations. Explicit analytic and numerical solutions illustrate the UV/IR relation and show how special sectors reduce to known Lorentzian AdS$_4$ black holes, while broader saddles correspond to non-extremal or non-Lorentzian configurations. These results extend holographic tests of the ABJM/AdS$_4$ duality beyond black hole entropy to a rich family of Euclidean saddles, offering new insights into supersymmetric localization, holographic renormalization, and the structure of quantum gravity path integrals.

Abstract

We find new asymptotically locally AdS$_4$ Euclidean supersymmetric solutions of the STU model in four-dimensional gauged supergravity. These "black saddles" have an $S^1\times Σ_{\mathfrak{g}}$ boundary at asymptotic infinity and cap off smoothly in the interior. The solutions can be uplifted to eleven dimensions and are holographically dual to the topologically twisted ABJM theory on $S^1\times Σ_{\mathfrak{g}}$. We show explicitly that the on-shell action of the black saddle solutions agrees exactly with the topologically twisted index of the ABJM theory in the planar limit for general values of the magnetic fluxes, flavor fugacities, and real masses. This agreement relies on a careful holographic renormalization analysis combined with a novel UV/IR holographic relation between supergravity parameters and field theory sources. The Euclidean black saddle solution space contains special points that can be Wick-rotated to regular Lorentzian supergravity backgrounds that correspond to the well-known supersymmetric dyonic AdS$_4$ black holes in the STU model.

Figures (7)

• Figure 1: A schematic depiction of the Euclidean black saddle solution \ref{['eq:univBS']} and the universal black hole \ref{['eq:univBH']}. In the limit $Q\to 0$ the periodicity of the $\tau$ coordinate becomes infinite and the IR region acquires the metric on $\mathbb{H}^2\times \Sigma_{\mathfrak{g}}$ which can then be analytically continued to the near-horizon AdS$_2 \times \Sigma_{\mathfrak{g}}$ metric of the black hole. Both classes of solutions are asymptotically locally $\mathbb{H}^4$ in the UV region with $S^1 \times \Sigma_{\mathfrak{g}}$ boundary.
• Figure 2: Numerical values of $\hat{\beta}$ for Case 1 as a function of $q_R$.
• Figure 3: Numerical values of the chemical potentials $\hat{\Delta}_1$ and $\hat{\Delta}_2$ normalized by $2\pi$ for Case 1 as a function of $q_R$. Also the combination $\hat{\Delta}_1+3\hat{\Delta}_2$ is shown which is expected to be equal to $2\pi$.
• Figure 4: Numerical values of $\hat{\Delta}_1$, $i\hat{\beta} \hat{\sigma}_1$ and the combination $\hat{\Delta}_1+i\hat{\beta}\hat{\sigma}_1$ for Case 1 as a function of $q_R$.
• Figure 5: Numerical values of $\hat{\Delta}_1$, $i\hat{\beta} \hat{\sigma}_1$ and the combination $\hat{\Delta}_1+i\hat{\beta}\hat{\sigma}_1$ for Case 2 as a function the IR value ${\tilde{z}}(0)$.