Thermodynamic stability from Lorentzian path integrals and codimension-two singularities (2501.08409v1)

Abstract: 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.

• The paper demonstrates that a black hole saddle's inclusion in the thermal partition function hinges on its stability against energy, angular momentum, and charge fluctuations.
• The study introduces and classifies codimension-two singularities with additional gauge shifts, thereby expanding the framework beyond traditional conical deficits.
• The research employs constrained saddles in 3D Einstein-Maxwell theory to quantify stable configurations, enhancing the predictive capacity of gravitational path integral models.

Thermodynamic Stability from Lorentzian Path Integrals and Codimension-Two Singularities

The paper by Hong Zhe (Vincent) Chen explores the computation of gravitational thermal partition functions via Lorentzian path integrals, focusing primarily on their application in the context of Einstein-Maxwell theory. Traditionally, the Euclidean path integral approach confronts obstacles due to the conformal factor problem, which is notably evaded in the Lorentzian framework. The primary investigation centers on whether the Lorentzian integration contour can be deformed to incorporate contributions from various black hole saddle points, contingent on their thermodynamic stability.

The author argues that the relevance of a particular black hole saddle point in the thermal partition function is intrinsically linked to its stability against fluctuations in energy, angular momentum, and charge. This involves a detailed examination of constrained saddles where terms such as area, angular momentum, and charge are held constant on a codimension-two surface, leading to the emergence of significant singularities.

The paper introduces a broader class of codimension-two singularities, expanding the theoretical framework to include these entities. The singularities are characterized not only by the traditional conical deficits but also by additional shifts along the surface and within the Maxwell gauge group. These shifts are acquired through a process of winding in metric-orthogonal and connection-horizontal directions near the surface.

To quantify these configurations, the paper proposes an action for singular setups, providing a mathematical basis for their incorporation into Lorentzian path integrals. The work specifically dedicates effort to three-dimensional spacetimes, which affords a simplified approach to handling the complexities associated with angular momentum.

The implications of this research stretch into both theoretical insights and practical advancements within the domain of quantum gravity. On a theoretical level, it identifies the conditions under which black hole configurations contribute to the partition function, thus influencing the thermodynamics and phase structure of gravitational systems. On a practical level, pinpointing stable configurations enhances the predictive capacity of gravitational path integral models, thereby contributing to a more comprehensive understanding of high-energy theoretical physics.

Future explorations may extend these methodologies to higher-dimensional spacetimes or assess additional fields' contributions, refining our models for quantum gravitational phenomena. Furthermore, the integration of such approaches could inform ongoing research endeavors in quantum mechanics and general relativity, fostering a nuanced comprehension of the underlying mechanics governing spacetime and its singularities.