A Brief Note on Complex AdS-Schwarzschild Black Holes

In the context of thermodynamics of asymptotically anti-de Sitter spaces, it is often stated that at very low temperatures, there is only one saddle point available-namely, thermal AdS-and hence this sole saddle dictates the low-temperature behavior. However, AdS-Schwarzschild black holes continue to exist at low temperatures as complex saddle points. We point out that the real part of the on-shell action of these complex black holes is smaller than that of thermal AdS at the lowest temperatures, in AdS$_5$ and higher dimensions. So, na\"ively, they should be the "dominant" saddles. This raises a puzzle: if these complex black holes were indeed the relevant saddle points, the physics of the bulk and that of the dual gauge theory would completely disagree at low temperatures. Using a mini-superspace approximation and contour arguments, we argue that these complex black holes do not actually contribute to the gravitational path integral, regardless of the value of their on-shell action. So the standard conclusion that thermal AdS is the correct saddle at the lowest temperatures continues to hold. We also comment on two related matters: whether the Kontsevich-Segal criterion is useful in this setting, and whether the unstable small black hole contributes to the path integral in the high-temperature phase.