Allowable Complex Black Holes in the Euclidean Gravitational Path Integral

TL;DR

This paper investigates the conditions under which complex black hole solutions are valid in the Euclidean gravitational path integral, linking violations of the KSW criterion to phase transitions in supersymmetric gauge theories.

Contribution

It applies the KSW criterion to complex black hole saddle points, revealing a connection between criterion violations and phase transitions in the superconformal index.

Findings

Violations of the KSW criterion occur where the index's statistical description fails.

Critical points align with phase transitions to 'grey galaxy' configurations.

The work links complex geometry conditions to physical phase transitions.

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.