Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics

TL;DR

The paper develops generalized asymptotically flat boundary conditions for 3D higher-spin gravity without a cosmological constant, preserving a higher-spin extension of the $BMS_{3}$ algebra and enabling thermodynamic analysis with chemical potentials. It shows these flat conditions arise as a careful $ abla o 0$ limit from AdS$_3$ boundary data via a permissible gauge, and provides a regularity framework based on holonomies that does not require a full matrix realization of the gauge group. The authors construct the higher-spin extension of locally flat cosmologies, derive the entropy from a boundary term in the Euclidean solid-torus topology, and obtain explicit results for pure gravity and higher-spin charged solutions, including a phase-parametrization of the thermodynamics and reality bounds. The work connects to related approaches and offers a solid path to extending these structures to spins $s\geq 2$, with implications for flat-space holography and the thermodynamics of higher-spin cosmologies.

Abstract

A generalized set of asymptotic conditions for higher spin gravity without cosmological constant in three spacetime dimensions is constructed. They include the most general temporal components of the gauge fields that manifestly preserve the original asymptotic higher spin extension of the BMS$_{3}$ algebra, with the same central charge. By virtue of a suitable permissible gauge choice, it is shown that this set can be directly recovered as a limit of the boundary conditions that have been recently constructed in the case of negative cosmological constant, whose asymptotic symmetries are spanned by two copies of the centrally-extended W$_{3}$ algebra. Since the generalized asymptotic conditions allow to incorporate chemical potentials conjugated to the higher spin charges, a higher spin extension of locally flat cosmological spacetimes becomes naturally included within the set. It is shown that their thermodynamic properties can be successfully obtained exclusively in terms of gauge fields and the topology of the Euclidean manifold, which is shown to be the one of a solid torus, but with reversed orientation as compared with one of the black holes. It is also worth highlighting that regularity of the fields can be ensured through a procedure that does not require an explicit matrix representation of the entire gauge group. In few words, we show that the temporal components of generalized dreibeins can be consistently gauged away, which partially fixes the chemical potentials, so that the remaining conditions can just be obtained by requiring the holonomy of the generalized spin connection along a thermal circle to be trivial. The extension of the generalized asymptotically flat behaviour to the case of spins $s\geq2$ is also discussed.

Figures (1)

• Figure 1: The sequence shows that the topology of the Euclidean cosmological spacetime coincides with the one of a black hole; i.e., it corresponds to $\mathbb{R}^{2}\times S^{1}$ (solid torus), but with reversed orientation (compare with fig. 1 of BHPTT). The cosmological horizon $r_{c}$ is located at the "south pole” of the $r-\tau$ surface, which asymptotically approaches to a conical defect at the tip of the drop, so that a regulator at $r=r_{0}$ has to be introduced. Noncontractible cycles then run along the circle $S^{1}$, being parametrized by the angle $\varphi$.