Asymptotic W-symmetries in three-dimensional higher-spin gauge theories

TL;DR

This work develops a systematic DS-reduction framework to compute asymptotic symmetries of three-dimensional bosonic higher-spin gauge theories in asymptotically AdS spaces, expressing the resulting algebras as W-algebras in a Virasoro-primary basis. It delivers closed formulas for the structure constants of the infinite-dimensional W_infty[lambda] and their finite N truncations W_N, clarifying how higher-spin generators organize as Virasoro primaries and how central terms arise. The paper further provides a quadratic basis for these algebras, with explicit basis transformations connecting highest-weight and quadratic presentations, and extends the construction to hs[lambda] ⊕ hs[lambda], including the metric-like interpretation of the fields. Together, these results solidify the link between DS reduction, higher-spin CS theories, and potential holographic duals, offering concrete tools for exploring W-algebras in AdS3 contexts.

Abstract

We discuss how to systematically compute the asymptotic symmetry algebras of generic three-dimensional bosonic higher-spin gauge theories in backgrounds that are asymptotically AdS. We apply these techniques to a one-parameter family of higher-spin gauge theories that can be considered as large N limits of SL(N) x SL(N) Chern-Simons theories, and we provide a closed formula for the structure constants of the resulting infinite-dimensional non-linear W-algebras. Along the way we provide a closed formula for the structure constants of all classical W_N algebras. In both examples the higher-spin generators of the W-algebras are Virasoro primaries. We eventually discuss how to relate our basis to a non-primary quadratic basis that was previously discussed in literature.