How to tame your (black hole) saddles: Lessons from the Lorentzian Gravitational Path Integral

TL;DR

This paper resolves divergence issues in the Lorentzian gravitational path integral for black holes by applying Picard-Lefshetz theory, showing only certain saddles contribute at finite temperature, and analyzing related BTZ black hole ensembles.

Contribution

It introduces a Lorentzian path integral approach with Picard-Lefshetz analysis to identify contributing saddles, ensuring convergence in black hole partition functions.

Findings

Only a finite subset of saddles contribute at finite temperature.

The sum over all saddles converges in the low temperature limit.

All saddles contribute in the BTZ black hole ensemble, and the sum converges.

Abstract

We resolve a puzzle associated with the spherically-symmetric sector of the AdS$_4$ Einstein-Maxwell partition function with inverse temperature $\beta$. Since charge is quantized, the semiclassical limit of the partition function is expected to be given by a sum over complex black hole solutions obtained by shifting the associated chemical potential $\mu$ by $\frac{2\pi i n}{e \beta}$ in terms of the relevant charge quantum $e$. However, the sum over all such saddles turns out to diverge at any finite value of $\beta$. We therefore consider a definition of this partition function as an integral over a space of metrics that are real and of Lorentz-signature up to the presence of certain conical singularities. A Picard-Lefshetz analysis shows that only a finite subset of the above saddles contribute to our integral at finite $\beta$, and thus that the sum over such saddles converges. The low temperature limit is nonetheless associated with a convergent sum over all saddles that (as $\beta \rightarrow \infty$) approach the usual large real Euclidean black holes. We also analyze the analogous partition function for the (uncharged) BTZ black hole in the ensemble defined by fixing an angular velocity $\Omega$ up to shifts by $\frac{2\pi i m}{s \beta}$, where $s=\frac{1}{2}$ or $s=1$ depending on the presence of absence of fermionic states. In this case, at all $\beta$ we find that all saddles contribute and that the sum over saddles converges. We also comment briefly on the apparent lack of utility of the so-called KSW condition in our context.