Complex Saddles of Charged-AdS Gravitational partition function

Abstract

In this paper, we consider the Euclidean partition function of uncharged and charged $AdS_{d+1}$ black hole geometries in canonical and grand canonical ensemble for $d\geq3$. It is seen that the partition function can be reduced to a one-dimensional integral, which can be investigated using methods of Picard-Lefschetz. The saddles of the system correspond to either naked-singular geometry, thermal-AdS, small-, intermediate- or large-sized black hole for different ranges of parameter space. These are solutions of Einstein's equation, which are dominant saddles of the partition function in various regimes of parameter space. A naive analysis of the partition function involving these saddles would lead to conflicts with the standard understanding of black hole thermodynamics and also with AdS/CFT. However, when the partition function is analysed using Picard-Lefschetz, it is seen that naked-singular geometries turn out to be irrelevant and therefore do not contribute. This also aligns well with the Cosmic Censorship hypothesis. Depending on the ensemble, saddles corresponding to negative specific heat are either small- or intermediate-sized black holes. Although they are relevant in the partition function but are sub-dominant. They drop out under homology averaging. Only saddles corresponding to non-negative specific heat contribute to the Euclidean partition function. Finally, we analyze the allowability of these complex geometries using the KSW criterion.