Higher-order chiral scalar from boundary reduction of 3d higher-spin gravity

TL;DR

The paper develops a covariant boundary reduction of three-dimensional higher-spin Chern-Simons gravity to produce Lorentz-covariant boundary actions for higher-order chiral scalars. At linear order, it generalizes the Floreanini-Jackiw and Alekseev-Shatashvili constructions to arbitrary spin and extends the analysis to non-linear asymptotic AdS conditions, including the $SL(3,\mathbb{R})$ HS case. Around BTZ-like backgrounds, the boundary theory becomes a gauged WZW model, with the gravity case reducing to AS action and the $N=3$ HS case yielding a nonlinear HS boundary theory. In the linearized HS sector with non-trivial HS charge ${\cal W}$, the edge modes form higher-derivative chiral scalars whose mode spectrum exhibits richer zero-mode structures tied to background data, revealing a spectrum of enhanced boundary symmetries and connections to ${\cal W}$-algebras.

Abstract

We use a recently proposed covariant procedure to reduce the Chern-Simons action of three-dimensional higher-spin gravity to the boundary, resulting in a Lorentz covariant action for higher-order chiral scalars. After gauge-fixing, we obtain a higher-derivative action generalizing the $s=1$ Floreanini-Jackiw and $s=2$ Alekseev-Shatashvili actions to arbitrary spin $s$. For simplicity, we treat the case of general spin at the linearized level, while the full non-linear asymptotic boundary conditions are presented in component form for the $SL(3,\mathbb R)$ case. Finally, we extend the spin-3 linearized analysis to a background with non-trivial higher-spin charge and show that it has a richer structure of zero modes.