Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity

TL;DR

The authors show that (2+1)-dimensional higher-spin AdS gravity, formulated as a Chern-Simons theory with hs(1,1), possesses an asymptotic symmetry algebra that is a nonlinear, centrally extended $W_ obreakd hypheninfty$ algebra. By imposing suitable boundary conditions and analyzing residual gauge transformations, they derive a closed Poisson-bracket structure among the HS charges, with the Virasoro subalgebra carrying the universal central charge $c = 3\ell/(2G)$. A explicit truncation to $W_3$ is shown to be consistent, yielding the classical nonlinear $W_3$ algebra with the same central charge, and the truncation corresponds to an $sl(3,R)$ subalgebra. The results reveal a rich infinite-dimensional symmetry at infinity, tightly constrained by hs(1,1)$ and the CS level $k$, with implications for quantum gravity, holography, and possible connections to integrable systems like the KP hierarchy.

Abstract

We investigate the asymptotic symmetry algebra of (2+1)-dimensional higher spin, anti-de Sitter gravity. We use the formulation of the theory as a Chern-Simons gauge theory based on the higher spin algebra hs(1,1). Expanding the gauge connection around asymptotically anti-de Sitter spacetime, we specify consistent boundary conditions on the higher spin gauge fields. We then study residual gauge transformation, the corresponding surface terms and their Poisson bracket algebra. We find that the asymptotic symmetry algebra is a nonlinearly deformed W(infinity) algebra with classical central charges. We discuss implications of our results to quantum gravity and to various situations in string theory.