Thermodynamic stability from Lorentzian path integrals and codimension-two singularities

It has previously been shown how the gravitational thermal partition function can be obtained from a Lorentzian path integral. Unlike the Euclidean case, the integration contour over Lorentzian metrics is not immediately ruled out by the conformal factor problem. One can then ask whether this contour can be deformed to pick up nontrivial contributions from various saddle points. In Einstein-Maxwell theory, we argue that the relevance of each black hole saddle to the thermal partition function depends on its thermodynamic stability against variations in energy, angular momentum, and charge. The argument involves consideration of constrained saddles where area and quantities associated with angular momentum and charge are fixed on a codimension-two surface. Consequently, this surface possesses not only a conical singularity, but two other types of singularities. The latter are characterized by shifts along the surface and along the Maxwell gauge group acquired as one winds around near the surface in a metric-orthogonal and connection-horizontal manner. We first study this enlarged class of codimension-two singularities in generality and propose an action for singular configurations. We then incorporate these configurations into the path integral calculation of the partition function, focusing on three-dimensional spacetimes to simplify the treatment of angular momentum.