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

Abstract

We resolve a puzzle associated with the spherically-symmetric sector of the AdS$_4$ Einstein-Maxwell partition function with inverse temperature $β$. 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 $μ$ by $\frac{2πi n}{e β}$ in terms of the relevant charge quantum $e$. However, the sum over all such saddles turns out to diverge at any finite value of $β$. 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 $β$, 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 $β\rightarrow \infty$) 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 $Ω$ up to shifts by $\frac{2πi m}{s β}$, where $s=\frac{1}{2}$ or $s=1$ depending on the presence of absence of fermionic states. In this case, at all $β$ 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.

Figures (10)

• Figure 1: The conformal diagram of an AdS-Schwarzschild black hole is shown with two marked surfaces differing by a time translation of magnitude $T$. Cutting along these surfaces defines a wedge. Identifying the two cut-surfaces of the wedge then yields a time-periodic spacetime with what we may call a Lorentzian conical singularity (red dot).
• Figure 2: A conformal diagram of an AdS-Reissner-Nordstrom black hole. The two green surfaces stretch from the inner horizon $\gamma$ to the right boundary. They are chosen to be related by a time translation of magnitude $T$. Cutting along these surfaces, and identifying the two surfaces yields a time-periodic spacetime with a Lorentzian conical singularity at $\gamma$. The periodic direction becomes null at the outer horizon and spacelike inside.
• Figure 3: The real part of $u$ for $\mu_0 = 2$ and $\frac{n}{qL}=3$, with darker and redder shading indicating more negative values of $\Re \, u$. The panels a,b,c,d show a sequence of plots with increasing $\beta L$. The green line represents the contour of integration $r_+ \in (0,\infty)$. The blue and red curves are constant phase curves for the corresponding saddles (blue and red dots, with blue denoting ${\mathcal{R}}_-$ and red denoting ${\mathcal{R}}_+$). The red steepest ascent line crosses the green contour in the final two panels, but not in the first two. This is consistent with the expectation that, for $qL = \frac{n}{3}< \sqrt{3}n$ and $\mu_0 \ge 1$, the red ($\mathcal{R}_+$) saddle should contribute at large $\beta$ but not at small $\beta$. In agreement with out analytic predictions, we also find the red saddle to approach the real line as $\beta \to \infty$. The blue ($\mathcal{R}_-$) saddle never contributes since the shading indicates that the blue line that crosses the real axis is the blue steepest descent line.
• Figure 4: The panels present $Z_3$ evaluated at $\hbar = 1, \mu_0 =2, n=3$ and $qL=1$. We believe that the sudden dip around $\beta \sim 17L$ is likely to be a result of working at finite $\hbar$, and that it would be replaced by a sharp phase transition in the semiclassical limit. In the left-top panel, we present the absolute value (blue) of the result of numerically integrating \ref{['Zn0']}. The right-top panel overplots the absolute value (red) of the boundary contribution \ref{['ref:boundary_contr']}. The left-bottom panel further overplots the absolute value (orange) of the contribution from the $\mathcal{R}_+$ saddle. Finely, the right-bottom panel sums the boundary contribution and the ${\mathcal{R}}_+$ saddle including one-loop corrections for both. Despite the fact that we work at $\hbar=1$, the result (magenta) completely covers the blue numerical curve, rendering the latter invisible. In particular, reproducing the dip required the correct phase difference between these two contributions. It thus serves as a check at the one-loop level.
• Figure 5: The panels show the real part of $u$ for $\mu_0 = 1$ and $\frac{n}{qL}=\frac{1}{2}$ using the same color coding as in Fig. \ref{['fig:mu2']}. The sequence $a,b,c,d$ again shows increasing $\beta L$. Since the red steepest ascent line crosses the green contour in each panel, the red (${\mathcal{R}}_+$) saddle always contributes. This is consistent with the expectation that for $qL = 2n> \sqrt{3}n$ and $\mu_0 \ge 1$, the red ($\mathcal{R}_+$) saddle should contribute at both small and large $\beta$. The blue ($\mathcal{R}_-$) saddle never contributes since the shading indicates that only its steepest descent line ever crosses the green contour.