Complex Saddles of Charged-AdS Gravitational partition function

TL;DR

This work uses Picard-Lefschetz theory to resolve a tension in Euclidean AdS black hole partition functions where dominant complex saddles (naked singularities) could naively dominate. By reducing $Z(\beta)$ to a one-dimensional integral and applying contour deformations, the authors show that naked singularities are irrelevant and that subdominant saddles drop out under homology averaging, restoring consistency with AdS/CFT and cosmic censorship. The analysis extends from uncharged Schwarzschild-AdS to charged AdS black holes in both grand canonical and canonical ensembles, revealing a robust pattern: only thermal AdS and thermodynamically stable large black holes contribute, while complex and negative-heat saddles are eliminated. The study also examines the KSW (weak) criterion for allowable complex geometries, finding it provides partial guidance but may warrant a stronger criterion for fully excluding unphysical saddles. Overall, the paper demonstrates that careful PL contouring and stability considerations yield physically sensible, dimension-spanning results in AdS thermodynamics.

Abstract

In this paper, we consider the Euclidean partition function of charged and uncharged AdS black hole geometries in (d+1)-dimensions. It is seen that the partition function can be reduced to a one-dimensional integral, which can be investigated using methods of Picard-Lefschetz. The saddles of the system correspond to either naked-singular geometry, thermal-AdS, small-, intermediate- or large-sized black hole for different ranges of parameter space. These are solutions of Einstein's equation, which are dominant saddles in the partition function in various regimes of parameter space. A naive analysis of the partition function involving these saddles would lead to conflicts with the standard understanding of black hole thermodynamics and also with AdS/CFT. However, when the partition function is analysed using Picard-Lefschetz, it is seen that naked-singular geometries don't contribute. The saddles corresponding to them are irrelevant, aligning well with the Cosmic Censorship hypothesis. Saddles corresponding to negative specific heat are either small- or intermediate-sized black holes. Although they are relevant in the partition function but are sub-dominant. They also drop out under homology averaging. Saddles corresponding to only non-negative specific heat contribute to the Euclidean partition function. Finally, we analyze the allowability of these complex geometries using the KSW criterion.

Figures (23)

• Figure 1: Steepest descent/ascent lines are plotted in black/red for $AdS_6$, for $\beta=5\beta_{\rm max}$, $l=1$. Thermal-AdS is at $r_+=0$ and $r_+^\pm$ are naked singularities. The purple arrow is the direction of integration cycle. Steepest ascent curves from these complex saddles don't intersect the real line and hence don't contribute to the partition function.
• Figure 2: Real part of $\mathcal{I}$ as a function of $\beta$ ($>\beta_{\rm max}$) at various saddles for $d=4$ and $d=7$. $Re[\mathcal{I}(r_+^\pm)]>0$ for complex saddles (naked-singular geometries) imply that for these values of $\beta$ naked singularities are dominant over the Thermal-AdS saddle.
• Figure 3: Steepest descent-ascent contour for $AdS_6$ in the high temperature regime ($\beta<\beta_{\rm max}$) with rotating $G=|G|e^{i\epsilon},|G|=1,l=1$. Steepest descent ($\mathcal{J}_\sigma$) curves are plotted in black, and steepest ascent ($\mathcal{K}_\sigma$) curves are plotted in red. The arrows indicate the direction of the integration cycle. Steepest ascent from all the saddles intersect the real line (for either sign) and hence contribute to the partition function. Note that due to the degeneracy of the ads saddle in $AdS_6$, there are three descent/ascent lines emanating from the saddle. However, only the branch lying on the positive axis participates in the Stokes jump.
• Figure 4: Steepest descent-ascent contour for $AdS_5$ in the degenerate case ($\beta=\beta_{\rm max}$) with real rotating $G=|G|e^{i\epsilon}$. Steepest descent ($\mathcal{J}_\sigma$) curves are plotted in black, and steepest ascent ($\mathcal{K}_\sigma$) curves are plotted in red. The arrows indicate the direction of the integration cycle. Note that due to the coalescing of saddles ($r_+^+=r_+^-$), there are extra descent/ascent lines emanating from the saddle (blue dot). However, the contribution from that branch drops out.
• Figure 5: Behaviour of specific heat ($C$) for $\beta>\beta_{\rm max}$ and for $\beta<\beta_{\rm max}$ at various saddles.