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

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.