⚛️ high-energy theory

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

This paper demonstrates that the Kontsevich-Segal-Witten criterion for the allowability of complex metrics in the gravitational path integral for AdS $_5$ black holes with two angular momenta is equivalent to the convergence conditions of the microscopic supersymmetric index, thereby extending previous equivalences found in simpler spacetime examples.

Original authors: Pietro Benetti Genolini, Oliver Janssen, Sameer Murthy

Published 2026-02-02

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CC BY 4.0

Original authors: Pietro Benetti Genolini, Oliver Janssen, Sameer Murthy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake the perfect cake. In the world of physics, specifically when trying to understand the universe at its smallest scales, scientists use a mathematical recipe called the "Gravitational Path Integral." Think of this recipe as a way to sum up every possible shape the universe could take to figure out how it actually behaves.

Usually, the ingredients in this recipe are "real" numbers, like the real distance between two points. But sometimes, to solve the math, physicists have to use "complex" numbers. In this context, "complex" doesn't mean "complicated"; it means numbers that have a real part and an imaginary part (like $3 + 4i$ ).

The problem is: Not all complex shapes make sense. If you use the wrong complex numbers in your recipe, the cake might turn into a black hole of nonsense, or the math might blow up. So, physicists need a rule to decide which complex shapes are "allowed" and which are "forbidden."

The "KSW" Rule: The Quality Control Inspector

In this paper, the authors are testing a specific rule called the Kontsevich–Segal–Witten (KSW) criterion. You can think of this rule as a strict quality control inspector. It checks every single point in a complex shape to ensure that the "energy" (or kinetic term) stays positive and doesn't go haywire. If a shape passes this test, it's an "allowable" saddle point—a valid candidate for the universe's shape.

The Mystery of the 5D Black Hole

The authors focused on a very specific, tricky ingredient: Supersymmetric black holes in a 5-dimensional space called AdS5.

Imagine these black holes as spinning tops with two different handles (angular momenta). In previous attempts to apply the KSW inspector to these spinning tops, there was a puzzle. When the two handles spun at different speeds, the inspector seemed to reject shapes that the "microscopic" recipe (the detailed, tiny-scale math of string theory) said should be allowed. It was like the inspector saying, "This cake is bad," while the recipe book said, "This cake is perfect."

The authors realized this was a mistake in how the inspector was being used. They fixed the way they applied the rule.

The Big Discovery: The Inspector and the Recipe Book Agree

Once they fixed the application of the KSW rule, they found something beautiful: The inspector and the recipe book finally agreed.

The Microscopic View: If you look at the black hole from the perspective of tiny quantum particles (the "microscopic" view), the math only works if certain conditions are met (like the temperature and spin being within specific ranges).
The KSW View: When they applied the KSW quality control rule to the complex geometry of the black hole, it rejected exactly the same shapes that the microscopic view rejected.

It turns out the KSW rule is perfectly tuned to the microscopic reality of these black holes. The "allowed" zone for the complex geometry is identical to the "allowed" zone for the quantum math.

How They Checked: From the Edge to the Center

To prove this, the authors looked at the black hole in two places:

The Edge (The Boundary): They checked the rule at the very edge of the space (the "conformal boundary"). Here, they could use pure math to prove that the KSW rule and the microscopic rules are twins. They match perfectly.
The Center (The Bulk): They then looked deep inside the black hole, near the event horizon. This part is too messy for pure math, so they used a computer to run thousands of simulations. They tested random shapes that passed the microscopic test to see if the KSW rule would catch any of them as "bad."
- The Result: The computer found zero failures. Every shape that the microscopic recipe said was good, the KSW inspector also said was good.

A Surprising Twist: The Rule Gets "Looser" Inside

They also noticed something interesting about how the rule behaves as you move from the edge to the center.

At the edge, the rule is very strict. It acts like a rigid fence.
As you move inside the black hole, the rule seems to get slightly "looser." The area of "forbidden" shapes shrinks as you go deeper.

Think of it like a security checkpoint at an airport. At the gate (the edge), the security is tightest. As you move further into the terminal (the bulk), the rules might relax slightly, but the people who made it past the gate are still the same people.

The Takeaway

This paper solves a puzzle about how we calculate the properties of black holes using complex math. It confirms that the KSW criterion is a reliable tool. It tells us that if a complex black hole shape is mathematically valid according to the tiny quantum rules, it will also pass the geometric "allowability" test.

The authors also provided a new, practical "algorithm" (a step-by-step checklist) for other scientists to use. Instead of doing difficult math to rotate coordinates, you can just plug the numbers into their checklist to see if a complex shape is allowed.

In short: The authors fixed a broken rule, proved that the rule works perfectly for 5D black holes, and showed that the "geometric" rules of the universe and the "quantum" rules of the universe are in perfect harmony.

Technical Summary: Allowable Complex Metrics and the Gravitational Index of AdS $_5$ Black Holes

Problem Statement
In the context of the Gravitational Path Integral (GPI), complex-valued metrics naturally arise as saddle points, particularly when calculating supersymmetric indices via analytic continuation (Wick rotation). However, not all complex metrics are physically admissible. The Kontsevich–Segal–Witten (KSW) criterion provides a necessary condition for a complex metric to be "allowable," ensuring the positivity of the real part of kinetic terms for fields propagating on the background.

A specific puzzle arose regarding supersymmetric black holes in asymptotically AdS $_5$ space carrying two unequal angular momenta. Previous investigations suggested a discrepancy: while microscopic constraints derived from the convergence of the superconformal index in the dual field theory allowed certain complex saddles, the KSW criterion appeared to rule them out. This paper identifies that this discrepancy stemmed from an implementation error in earlier work. The primary objective is to rigorously test the KSW criterion against the microscopic constraints for these AdS $_5$ black holes to determine if the two frameworks yield identical allowed regions in parameter space.

Methodology
The authors analyze the complex supersymmetric solutions of minimal gauged supergravity in five dimensions, which describe rotating, electrically charged black holes. The methodology proceeds through several stages:

Geometric Setup and Analytic Continuation: The authors review the Lorentzian supersymmetric black hole solutions and their complex deformations via Wick rotation ( $t = -i\beta t_E$ ). The solutions are parameterized by three real numbers $(a, r_+, r_*)$ , where $a$ relates to angular momentum, $r_+$ to the horizon radius, and $r_*$ to the extremal horizon. The complex metric and gauge field are defined on $\mathbb{R}^2 \times S^3$ .
Microscopic Constraints: The authors derive the convergence conditions for the superconformal index in the dual 4d $\mathcal{N}=1$ SCFT. These conditions require the real part of the inverse temperature ( $\text{Re}\,\beta > 0$ ) and the imaginary parts of the chemical potentials ( $\text{Im}\,\sigma > 0, \text{Im}\,\tau > 0$ ) to ensure the trace over the BPS subspace converges. These define a region in parameter space denoted as micro.
KSW Criterion Implementation:
- Algorithmic Reformulation: To facilitate numerical testing, the authors derive a practical algorithm based on the eigenvalues of a specific matrix constructed from the metric. A metric $g$ is KSW-allowable if, at every point, $\det g \notin \mathbb{R}^-$ , the matrix $A = \text{Re}(\sqrt{\det g} g^{-1})$ is positive definite, and the sum of the absolute values of the arguments of the eigenvalues of $gA$ is less than $\pi$ .
- Boundary Analysis: The criterion is applied analytically to the conformal boundary using the Fefferman–Graham expansion. The authors demonstrate that at the boundary, the KSW conditions are equivalent to the microscopic convergence conditions.
- Bulk Analysis: Since an analytical proof for the entire bulk is intractable, the authors perform extensive numerical evaluations. They sample points in the parameter space $(a, r_+, r_*)$ that satisfy the microscopic constraints and test the KSW conditions at various radial depths ( $r$ ) and angular positions ( $\theta$ ).

Key Contributions and Results

Resolution of the Discrepancy: The paper corrects a previous implementation error regarding unequal angular momenta. Upon proper application of the KSW criterion, the authors show that the allowed region for the AdS $_5$ black hole is exactly the same as the region defined by the microscopic convergence of the superconformal index.
Equivalence of Criteria: The authors establish the equality:
$\text{KSW} = \text{micro}$
for the supersymmetric index in AdS $_5$ . This extends previous findings in 4d flat space, 4d AdS, and other simpler examples, suggesting a robust agreement between gravitational allowability and microscopic consistency for supersymmetric indices.
Boundary vs. Bulk Behavior:
- Boundary: The KSW criterion at the conformal boundary is shown to be equivalent to the convergence of the microscopic trace. The authors clarify that while the KS criterion applied to a generic 4d CFT on a complex background imposes stricter conditions ( $\frac{1}{\pi}\text{Re}\,\beta > \text{Im}\,\sigma, \text{Im}\,\tau$ ), the specific geometric relations of the AdS $_5$ bulk saddle (linking $\beta, \sigma, \tau$ ) render these stricter conditions redundant, leaving only the microscopic constraints.
- Bulk: Numerical results indicate that the KSW constraints are strongest at the conformal boundary and become weaker as one moves into the bulk (towards the horizon). The region of parameter space ruled out by the KSW criterion shrinks as the radial coordinate $r$ decreases. While the authors could not prove this monotonicity analytically, their numerical data supports the proposition that $KSW_{R_2} \subset KSW_{R_1}$ for $R_2 > R_1$ .
Practical Algorithm: The paper provides a concrete algorithm (Appendix A) for checking KSW allowability using eigenvalues of the matrix $gA$, avoiding the need to diagonalize the metric by congruence, which is computationally advantageous.

Significance and Claims
The paper claims that the KSW criterion works "precisely" for the gravitational supersymmetric index in AdS $_5$ , matching the independent constraints derived from the dual microscopic theory. This result reinforces the validity of the KSW criterion as a selection rule for saddle points in the gravitational path integral.

The authors note that while this agreement is observed in a variety of supersymmetric settings (AdS and flat space), it is not yet known if it holds for the general gravitational path integral without supersymmetry. They suggest that the existence of a Killing spinor in these geometries might be the underlying reason for this agreement, though a definitive explanation is not provided. The work also highlights that the KSW condition weakens in the deep interior of the black hole, a phenomenon that currently lacks a simple analytic explanation.

The paper does not propose new experimental applications or future extensions beyond suggesting that the KSW criterion could be tested in more complex gauged supergravity models (e.g., $U(1)^3$ ) or with complex gauge fields, and that the relationship between the KSW criterion and D-brane stability criteria warrants further investigation.