On the Explicit Asymptotic $\mathcal{W}_5$ Symmetry of 3D Chern-Simons Higher Spin $AdS_3$ Gravity

Abstract

In this paper, we explicitly construct an asymptotic $\mathcal{W}_5$ symmetry algebra of the three-dimensional anti-de Sitter $(AdS_3)$ higher spin gravity. We use an $\mathfrak{sl}(5,R) \oplus \mathfrak{sl}(5,R)$ Lie algebra valued Chern-Simons gauge theory with a negative cosmological constant and its asymptotic symmetry algebra is explicitly calculated as two copies of the classical $\mathcal{W}_5$ algebra with central charge c. Our results can be interpreted as a spin $5$ extension of $AdS_3$ gravity and a proof of how the higher spin Ward identities and as well as the asymptotic $\mathcal{W}_5$ symmetry algebra are derived from the higher spin bulk field equations of motion. This higher spin asymptotic $\mathcal{W}_5$ symmetry algebra contains a finite number of conformal primary spin s: $s\,=\,2,\,3,\,4,\,5.$ We also indicated how to introduce chemical potentials and holonomy conditions associated with these higher spin charges in $AdS_3$ higher spin gravity in a manner that it preserves the asymptotic symmetry algebra.