Black Hole Calculations Reveal Limits of Statistical Physics and ‘grey Galaxy’ States

Scientists are increasingly utilising the Euclidean Gravitational Path Integral to model black hole physics, but its limits regarding complex metrics remain poorly understood. Vineeth Krishna and Finn Larsen, both from the Leinweber Institute for Theoretical Physics at the University of Michigan, investigate the applicability of the Kontsevich-Segal-Witten (KSW) criterion to determine allowable complex metrics within this framework. Their research reveals that black hole saddle points violate the KSW criterion at the same points where established statistical descriptions of the super-conformal index of SYM with unequal angular momenta fail, suggesting a fundamental connection between mathematical admissibility and physical breakdown. This correspondence pinpoints a critical point coinciding with the formation of two-component “grey galaxy” configurations, offering new insights into the micro-canonical ensemble and the nature of black hole entropy.

This work rigorously examines the Kontsevich-Segal-Witten (KSW) criterion, a proposal for determining which complex metrics can be legitimately included in the gravitational path integral.

Researchers discovered that for the super-conformal index of N = 4 Supersymmetric Yang-Mills (SYM) with unequal angular momenta, violations of the KSW criterion precisely coincide with the breakdown of the statistical description of the index. By leveraging the AdS/CFT correspondence, the research connects bulk gravitational calculations with computations in the strongly interacting boundary conformal field theory.
The superconformal index, a protected quantity in superconformal field theories, serves as a robust tool for constraining the rules governing the gravitational path integral, particularly in regions where the index converges. Specifically, the investigation focuses on non-extremal analytic continuations of BPS black hole solutions, which necessarily introduce complex spacetime metrics.

Researchers developed a practical method for implementing the KSW criterion to determine which of these complex metrics are admissible, demonstrating that the criterion accurately predicts the region of convergence for the superconformal index. The findings are nontrivial for N = 4 SYM when angular velocities are unequal but electric charges remain consistent, revealing a nuanced relationship between the KSW conditions and the physical validity of the gravitational path integral.

Furthermore, this work corrects a previous calculation regarding the KSW criteria for unequal angular momenta, confirming the insufficiency of these conditions in regions where the boundary superconformal index diverges, mirroring results obtained with complex Euclidean AdS4 black hole saddles carrying equal charges. The study then developed a practical method for implementing the KSW criterion, enabling the determination of admissible complex metrics crucial for evaluating the path integral.

This involved analysing the partition function, formally defined as a path integral over spacetime metrics and matter fields weighted by the Euclidean gravitational action, and approximating it via a sum over saddle points in the semiclassical limit. The on-shell Euclidean action was evaluated at each saddle point, solutions to the equations of motion satisfying imposed boundary conditions, with the leading contribution arising from the saddle point minimizing the real part of the action.

Furthermore, the Superconformal Index, defined for N = 4 supersymmetric Yang-Mills theory on S3 × R, was employed to connect boundary CFT observables with bulk gravitational computations. The index, expressed as a trace over the CFT Hilbert space, incorporates chemical potentials associated with angular momenta and U(1) charges, and a constraint ensuring contributions only from BPS states. Investigations focused on the super-conformal index of SYM with unequal angular momenta, revealing violations of the KSW criterion coinciding with the breakdown of the statistical description of the index.

These violations specifically occur at a critical point corresponding to two-component “grey galaxy” configurations observed in the micro-canonical ensemble. Analysis of AdS5 Black Holes with Unequal Angular Momenta and Lorentzian Black Hole Solutions formed a basis for understanding the Euclidean saddle points relevant to the index calculation.

Stability assessments of these Euclidean saddle points were conducted for various values of the parameter ‘p’, with detailed examination under the KSW condition for p = 0, p = 1, and p greater than or equal to 2. The KSW conditions were further investigated within the bulk of spacetime to determine their influence on admissible metrics.

The study demonstrates that the KSW criteria are satisfied if and only if the superconformal index is well-defined, establishing a direct link between metric admissibility and index convergence. Physical regions were determined for N = 4 Supersymmetric Yang-Mills theory with unequal angular velocities but equal electric charges.

In complex Euclidean AdS4 black hole saddles carrying two pairs of equal charges, the KSW conditions proved insufficient, failing to exclude complex saddles in regions where the boundary SCI diverges. Researchers anticipate similar insufficiency for AdS5 black holes with unequal electric charges. Investigations into super-conformal field theories reveal that violations of the Kontsevich-Segal-Witten (KSW) criterion, which determines admissible complex metrics, coincide with the breakdown of the statistical description of the superconformal index.

This critical point corresponds to the formation of two-component “grey galaxy” configurations when analysing the system using the micro-canonical ensemble. The findings demonstrate a holographic consistency check for the gravitational path integral, linking bulk geometry to boundary field theory calculations.

Specifically, the research validates a rule for selecting physical saddle points in the path integral by requiring agreement with the convergence region of the superconformal index. Future research may explore the broader implications of these findings for understanding the rules governing gravitational path integrals and resolving the black hole information paradox.

Further investigation into the nature of “grey galaxy” configurations and their role in the statistical description of the superconformal index could also prove fruitful. These results offer a refined understanding of the interplay between gravity and quantum field theory, potentially guiding the development of a more complete theory of quantum gravity.