On the Explicit Asymptotic ${\mathcal{W}}_5$ Symmetry of 3D Chern-Simons Higher Spin $Ad{S}_3$ Gravity

In this paper, we explicitly construct an asymptotic ${\mathcal{W}}_5$ symmetry algebra of the three-dimensional anti-de Sitter $(Ad{S}_3)$ higher spin gravity. We use an $\mathfrak{s}\mathfrak{l}(5,R)\oplus \mathfrak{s}\mathfrak{l}(5,R)$ 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 $Ad{S}_3$ 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 $Ad{S}_3$ higher spin gravity in a manner that it preserves the asymptotic symmetry algebra.