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

TL;DR

This paper explores the allowability of complex metrics in the gravitational path integral for AdS$_5$ black holes, establishing a link between the KSW criterion and the convergence of the supersymmetric index, with practical computational methods.

Contribution

It demonstrates the equivalence of the KSW allowability criterion and the convergence conditions of the supersymmetric index for AdS$_5$ black holes with two angular momenta, and provides an algorithm for implementation.

Findings

KSW criterion matches the index convergence conditions for AdS$_5$ black holes.

Practical eigenvalue-based algorithm for the KSW criterion.

Extension of previous results to more complex black hole configurations.

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.