Allowable Complex Black Holes in the Euclidean Gravitational Path Integral

TL;DR

This work tests the Kontsevich-Segal-Witten (KSW) criterion for admissible complex metrics against the Euclidean Gravitational Path Integral realization of the superconformal index in ${ m exists N}=4$ SYM with unequal angular momenta. By analyzing analytic continuations of $AdS_5$ black holes, the authors show that complex Euclidean saddles violate the KSW criterion precisely in the region where the SCI converges fails, and that the critical point corresponds to a phase transition to a two-component Grey Galaxy configuration in the microcanonical ensemble. The results establish a direct link between bulk geometric admissibility and boundary index convergence, suggesting KSW as a diagnostic tool for selecting physical saddles and highlighting the role of multi-component phases in the gravitational path integral. The study further discusses limitations, potential extensions to other charges and dimensions, and the connection to alternative criteria for complex metrics in Euclidean quantum gravity.

Abstract

The Euclidean Gravitational Path Integral has proven remarkably effective in the quantum regime of black hole physics. In this work, we examine the applicability of the Kontsevich-Segal-Witten (KSW) criterion for admissible complex metrics in the context of the Euclidean Gravitational Path Integral. We find that, for the super-conformal index of ${\cal N}=4$ SYM with unequal angular momenta, the black hole saddle points violate the KSW criterion precisely where the statistical description of the index breaks down. The corresponding critical point coincides with a phase transition into two-component ``grey galaxy'' configurations in the micro-canonical ensemble.

Figures (2)

• Figure 1: The $(a, r_+)$ cross sections of the parameter space at constant $r_*=0.1$. The green shaded region represents the parameter space where the $p=1$ KSW criterion is satisfied in the asymptotic AdS$_5$ region of the complex metric. The plots from the left to right take $\vartheta=0,\pi/4,\pi/2$. The KSW condition is satisfied for all $\vartheta$ inside the red curves. These are precisely the boundaries of the region of convergence of the superconformal index. The plots were obtained using the function RegionPlot in Mathematica with an initial grid of $2500$ points and with a MaxRecursion of 2.
• Figure 2: The plots show $(a, r_+)$ cross sections of the parameter space at constant $r_*=0.1$. The green region represents the region of parameter space where the corresponding black hole saddle point satisfies all $p\geq0$ KSW conditions. The red curves are the boundaries of the region of convergence of the index. The columns from the left to right show the validity of the KSW condition at $\vartheta=0,\pi/4,\pi/2$. The rows from top to bottom show the validity of the KSW condition at $r=0.5,2,4,\infty$. We observe that the strongest constraints are obtained at $r=\infty$ and $\vartheta = 0,\pi/2$. The plots were obtained using the function RegionPlot in Mathematica with an initial grid of $2500$ points and with a MaxRecursion of $2$.