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.
Highlights
Black hole solution in [57, 58] reduces to a higher spin extension of locally flat cosmological spacetimes [64,65,66,67,68]
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 W3 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
Summary
Higher spin gravity in three spacetime dimensions is remarkably simpler than its higherdimensional counterparts [72, 73]. We explain how the extended asymptotically flat structure described above can be recovered from a suitable vanishing cosmological constant limit of the asymptotic conditions recently constructed in [57, 58], for the case that includes black holes endowed with global higher spin charges. As explained in [57, 58], since all the chemical potentials are manifestly incorporated along the temporal components of the gauge fields, the analysis can be carried out for a fixed range of the angular coordinates of the torus, i.e., we assume that 0 < τ ≤ 1, and 0 < φ ≤ 2π This is useful in the case of higher spin gravity, since the torus clearly has not enough room to accommodate all the chemical potentials in the range of coordinates. As a warming up exercise, this is first performed in the case of pure gravity, and we show how the analysis extends once the higher spin charges and their corresponding chemical potentials are switched on
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