On the explicit asymptotic W5 symmetry of 3D Chern-Simons higher spin AdS₃ gravity

In this paper, we explicitly construct an asymptotic W5 symmetry algebra of the three-dimensional anti-de Sitter (AdS₃) higher spin gravity. We use an sl(5,R)⊕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 W5 algebra with central charge c. Our results can be interpreted as a spin 5 extension of AdS₃ gravity and a proof of how the higher spin Ward identities and the asymptotic W5 symmetry algebra is derived from the higher spin bulk field equations of motion. This higher spin asymptotic W5 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 AdS₃ higher spin gravity in a manner that it preserves the asymptotic symmetry algebra.