Allowable complex metrics and the gravitational index of AdS$_5$ black holes

TL;DR

The paper analyzes the Kontsevich–Segal–Witten criterion for the allowability of complex metrics in the gravitational path integral describing the supersymmetric index of AdS$_5$ black holes with two angular momenta. It develops an eigenvalue-based algorithm to test KSW stability and demonstrates that, at the conformal boundary, the KSW conditions are equivalent to the microscopic convergence constraints on the BPS trace; extending this into the bulk via the Fefferman–Graham expansion and numerical study shows that the KSW constraints do not enlarge the allowed region beyond the microscopic one. The authors prove that BKSW coincides with the microscopic region and provide strong numerical evidence that the full KSW criterion, when applied to these complex AdS$_5$ saddles, yields the same allowed region as the microscopic index, i.e., $ extsf{KSW}= extsf{micro}$. This builds on prior results in AdS and flat space and clarifies the interplay between gravitational allowability and microscopic convergence for a holographic supersymmetric index. The work also offers a practical computational approach for implementing KSW in holographic contexts and highlights the tension between boundary and bulk conditions in complex geometries.

Abstract

We discuss the Kontsevich-Segal-Witten criterion for the allowability of complex metrics, in the context of the gravitational path integral that calculates the supersymmetric index. We focus on the saddle points that capture the contribution of supersymmetric black holes in AdS$_5$ space. We show that, for such black holes with two independent angular momenta, the conditions imposed on the corresponding saddle point by the KSW criterion are equivalent to the ones arising from the convergence of the microscopic trace form of the supersymmetric index. This result adds to previous results establishing such an equivalence in other, simpler examples of the gravitational index in AdS space and flat space. Along the way, we give a practical algorithm for implementing the KSW criterion in terms of eigenvalues of certain matrices.

Figures (4)

• Figure 1: Plot of geometric and microscopic constraints (cross-section at $r_+=r_\star$). The figure shows the the $(r_\star,a)$-plane at the extremal point $r_+=r_\star$. The region bounded by the horizontal green line $a=1$, the blue line $a = (r_\star^2 -1)/2$ ($b=1$), and the vertical axis contains all the points allowed by the geometric constraints given in \ref{['eq:Geometric_Constraint_Degeneracy_v2']}. The orange region is where $\text{Im} \,\sigma_g >0$ and $\operatorname{Im} \tau_g >0$. The red region is where $\text{Im} \,\sigma_g >0$ but $\text{Im} \, \tau_g <0$. The yellow region is where $\text{Im} \, \tau_g >0$ but $\text{Im} \, \sigma_g <0$. The red-orange separator $\operatorname{Im} \sigma_g = 0$ is given by $a=X(r_\star)$. The yellow-orange separator $\operatorname{Im} \tau_g =0$ is given by $a=Y(r_\star)$.
• Figure 2: Plot of geometric and microscopic constraints (cross-section at $r_+=r_\star$). The figure shows the the $(b,a)$-plane at the extremal point $r_+=r_\star$. The boundaries of the region are at $r_\star = \sqrt{a+b+ab}=0$, $a=1$ and $b=1$. The color coding of the regions, and the colors and styles of the various curves are the same as in Figure \ref{['fig:5d_Imtau_BPS_v2']}.
• Figure 3: Plot of geometric and microscopic constraints (cross-section at fixed $r_\star$). The figure shows the the $(a,r_+)$-plane at fixed $r_\star = 0.01, 0.1, 0.3, 0.5$. The colored region (all colors) represents the region allowed by geometric constraints given in \ref{['eq:Geometric_Constraint_Degeneracy_v2']}. The microscopic constraints described in \ref{['eq:5d_AdSKN_RegionsPositivity']} divide the region into three parts, as in Figure \ref{['fig:5d_Imtau_BPS_v2']} with the same color coding: red = $\text{Im} \, \sigma_g >0$ and $\text{Im} \, \tau_g < 0$, yellow = $\text{Im} \, \tau_g >0$ and $\text{Im} \, \sigma_g < 0$, orange = both $\text{Im} \, \sigma_g , \, \text{Im} \, \tau_g >0$.
• Figure 4: Plot of microscopic constraints, geometric constraints and numerical evaluation of the KSW criteria. The figure shows the regions $\textsf{KSW}_R$ defined in \ref{['eq:KSWR_def']} for $R=nr_+$ and $n= 4, 5, 6, 8$. At fixed $r_\star = 1/6$, $a$ and $r_+$ are uniform-randomly picked within the geometric region, and the algorithm around \ref{['KSWeigenvalueEq']} is run for all $\theta \in [0,\pi/2]$. A green dot denotes that the algorithm is passed, and thus the metric belongs to the family $\textsf{KSW}_{R}$ for the specific choice of $R$. In contrast, a black dot indicates that the KSW algorithm was not passed: there was a $\theta \in [0,\pi/2]$ for which \ref{['KSWeigenvalueEq']} did not hold. We remark that the allowed region appears to shrink as $R$ increases, approaching the microscopic convergence region \ref{['microregion']} as $R \to \infty$.