⚛️ high-energy theory

Allowable complex saddles with real charges in the gravitational index

This paper introduces a new class of supersymmetric, non-extremal complexified AdS black holes with real charges and multiple complex horizons, demonstrating that while their asymptotic behavior aligns with the convergence of the dual superconformal index, the stricter Kontsevich-Segal-Witten criterion for the full bulk geometry excludes many saddles that would otherwise be considered admissible.

Original authors: Fabio Billiato, Gianmarco Esposto

Published 2026-09-30

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CC BY 4.0

Original authors: Fabio Billiato, Gianmarco Esposto

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

In the quest to understand the universe at its smallest scales, physicists often rely on a powerful idea called the gravitational path integral. Imagine trying to predict the behavior of a complex system by considering every possible way it could happen, then adding up the results to find the most likely outcome. In the realm of gravity, this means summing over every possible shape a universe or a black hole could take. For decades, scientists have used this method to study black holes, particularly those floating in a space with a negative curvature known as anti-de Sitter space. These environments are special because they act as a mirror to a different kind of physics called quantum field theory, allowing researchers to translate difficult gravitational problems into more manageable calculations on the boundary of the space.

A key tool in this translation is the superconformal index, a specific calculation in the quantum theory that counts certain stable states. For this calculation to work, the numbers used to describe the black hole must stay within a specific range, much like a radio signal that only works if tuned to the right frequency. On the gravity side, finding the right shapes to plug into the path integral has traditionally required stretching the rules of reality, allowing the black holes to become complex and non-extremal. This means the mathematical descriptions of these objects involve imaginary numbers and do not look like the solid, real black holes we might picture. The challenge has been figuring out which of these strange, complex shapes are actually valid contributors to the physics and which are just mathematical ghosts that should be ignored.

Two researchers from the University of Padua in Italy have taken a fresh look at this problem by proposing a new way to construct these complex black holes. Instead of making the charges of the black hole imaginary, as was done in previous studies, they kept the physical charges real and allowed the location of the black hole's horizon to become complex. This subtle shift led to a landscape of solutions with multiple horizons, some of which are real and others that exist only as complex mathematical points. By applying a rigorous test known as the Kontsevich-Segal-Witten criterion, which checks if a geometry can support a stable quantum theory, the team discovered that many of the previously considered solutions are actually forbidden.

The study reveals that while the mathematical conditions for the quantum index to converge are met for a wide range of these complex black holes, the gravitational rules are far more strict. In both four and five dimensions, the researchers found that the complex horizons that were thought to be valid candidates for the path integral are often ruled out by the deeper structure of the space itself. Specifically, the test showed that "virtual" horizons, which are purely complex and never become real, cannot be part of the physical description. More surprisingly, even some of the outer horizons that satisfied the boundary conditions failed the test when the entire interior of the black hole was examined. This suggests that the space between the center and the edge of the black hole imposes constraints that the boundary alone does not see.

The findings indicate that the standard way of checking if a complex black hole is allowed might be too loose when applied to supersymmetric systems. The researchers suggest that the current test, which checks if the geometry supports a generic quantum field, might be missing the special cancellations that happen between different types of particles in a supersymmetric setup. This mismatch means that some solutions that look good on the surface are actually inadmissible when the full complexity of the interior is taken into account. The work does not claim to have solved the entire puzzle of the gravitational path integral, but it provides a clearer map of which complex geometries are safe to use and which lead to inconsistencies. By identifying these forbidden regions, the study helps refine the tools physicists use to connect the gravity of black holes with the quantum mechanics of the universe.

Technical Summary: Allowable Complex Saddles with Real Charges in the Gravitational Index

Problem Statement
The gravitational path integral (GPI) in the semiclassical approximation is evaluated as a sum over saddle points compatible with boundary conditions. For asymptotically Anti-de Sitter (AdS) spacetimes, the AdS/CFT correspondence relates the gravitational index to the superconformal index (SCI) of the dual field theory. While the SCI is well-defined, reproducing it from the gravity side requires extending the integration contour to a complexified space of metrics and matter fields.

Previous analyses typically constructed complex supersymmetric saddles by complexifying solution parameters (such as mass or charge) while keeping spacetime coordinates real, resulting in complex conserved charges. This paper addresses the alternative prescription: constructing complex saddles that preserve real conserved charges (mass, angular momentum, electric charge) while allowing the radial coordinate and horizon positions to become complex. The central problem is determining which of these new complex saddles are "allowable" (physically valid) contributions to the GPI. Specifically, the authors investigate whether the Kontsevich-Segal-Witten (KSW) criterion, which ensures the well-definedness of quantum field theories on the background, aligns with the convergence conditions of the dual superconformal index.

Methodology
The authors analyze charged, rotating black holes in both AdS $_4$ and AdS $_5$ within minimal gauged supergravity.

Complexification Scheme: Instead of complexifying parameters to keep the horizon real, the authors keep the parameters $a$ (rotation), $m$ (mass), and $\delta$ (charge parameter) real. This forces the roots of the blackening function $\Delta_r(r)$ (the horizons) to become complex. They distinguish between:
- Outer horizons: Roots that can become real and positive in the extremal limit.
- Virtual horizons: Roots that remain complex for all physical parameters.
  The radial integration contour is defined as a path in the complex $r$ -plane starting from a complex horizon $r_+$ and extending to the asymptotic boundary along the real axis.
Supersymmetry Constraints: The authors impose supersymmetry conditions on the solution. A key finding (proven in Appendix A) is that requiring real conserved charges in a supersymmetric, non-extremal setup forces the reduced chemical potentials ( $\omega, \phi$ ) to lie strictly on the BPS locus (the one-dimensional curve defined by BPS solutions), even though the solutions themselves are non-extremal.
Allowability Criteria:
- Index Convergence: The authors derive the conditions under which the dual field theory index converges (e.g., $\text{Re}(\beta) > 0$ , $\text{Re}(\omega) < 0$ ).
- KSW Criterion: They apply the KSW criterion, which requires that the real parts of the kinetic terms for any probe field be positive. This is tested in three regimes:
  - Asymptotic Region: Checking the boundary metric.
  - Near-Horizon Region: Checking the metric near the complex horizon, dependent on the contour's tangent direction $\kappa$ .
  - Bulk Region: Checking the metric throughout the interior, requiring the existence of a continuous contour where the criterion holds at every point.

Key Contributions and Results

AdS $_4$ Analysis:
- Virtual Horizons: Both the index convergence conditions and the KSW criterion (asymptotic and bulk) exclude saddles associated with virtual horizons.
- Outer Horizons: The asymptotic KSW criterion agrees with the index convergence conditions. However, the bulk KSW analysis is more stringent. It excludes a subset of saddles that satisfy the asymptotic conditions and index convergence. Specifically, the bulk criterion removes solutions lying too close to the boundaries of the parameter space (near singular limits like $a=1$ or $a=0$ ).
- Contour Existence: The authors demonstrate that even if a horizon and the asymptotic region are KSW-allowable, the intermediate bulk may contain "forbidden" regions that prevent a continuous admissible contour from connecting the horizon to the boundary.
AdS $_5$ Analysis:
- Virtual Horizons: Saddles associated with purely imaginary virtual horizons are excluded. They possess purely imaginary inverse temperatures ( $\beta \in i\mathbb{R}$ ), leading to Lorentzian signatures at the boundary and closed timelike curves, rendering them inadmissible.
- Asymptotic Discrepancy: Unlike the AdS $_4$ case and previous studies of standard complexifications, the asymptotic KSW criterion in AdS $_5$ is more restrictive than the index convergence condition when the angular momenta are unequal ( $J_1 \neq J_2$ ). It excludes saddles whose chemical potentials lie within the index-convergence domain.
- Bulk Constraints: The bulk KSW analysis imposes the strongest constraints, removing additional saddles near the boundaries of the horizon domain (associated with singular solutions where rotation parameters approach unity).

Significance and Claims
The paper claims that the formulation of the KSW criterion used (which tests a generic probe field theory on a fixed background) is too restrictive for supersymmetric setups involving cancellations between bosonic and fermionic states.

Mismatch in AdS $_5$ : The fact that the asymptotic KSW criterion excludes saddles that the field-theory index accepts suggests that the KSW criterion, in its current form, fails to capture the specific constraints of the supersymmetric sector (BPS sector). The authors interpret this as an indication that the criterion needs refinement to account for the supersymmetric structure responsible for the index's convergence.
Bulk vs. Boundary: The results highlight that boundary conditions alone are insufficient to determine the validity of a saddle. The bulk geometry must also support a valid contour. The presence of forbidden bulk regions suggests that the transition from allowable to non-allowable saddles might be associated with phase transitions or a change in the dominant saddle of the path integral.
Real Charges Prescription: The study establishes that the "real charges" complexification explores a distinct section of the supersymmetric solution space compared to the standard "complex charges" approach. While both yield complex saddles, the "real charges" approach forces chemical potentials onto the BPS locus, a feature not present in the standard prescription.

The authors conclude that while the KSW criterion successfully rules out clearly pathological saddles (like virtual horizons), its application to regular complex supersymmetric saddles reveals tensions with field-theory expectations, particularly in five dimensions, pointing to the need for a more nuanced understanding of allowability in supersymmetric gravitational path integrals.

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