$W(0,b)$ algebra and the dual theory of 3D asymptotically flat higher spin gravity

TL;DR

This paper explores the structure of $W(0,b)$ algebras in 3D flat higher spin gravity, showing that only specific algebras admit a Chern-Simons formulation and constructing a dual boundary theory.

Contribution

It demonstrates the presence of $W(0,-1)$ and $W(0,-2)$ subalgebras in the asymptotic symmetry algebra of 3D flat higher spin gravity and constructs a dual boundary field theory.

Findings

Only $W(0,-1)$ admits a non-degenerate bilinear form for Chern-Simons formulation.

The asymptotic symmetry algebra contains $W(0,-1)$ and $W(0,-2)$ as subalgebras.

A dual boundary field theory is constructed using Chern-Simons/Wess-Zumino-Witten correspondence.

Abstract

BMS algebra in three spacetime dimensions can be deformed into a two parameter family of algebra known as $W(a,b)$ algebra. For $a=0$, we show that other than $W(0,-1)$, no other $W(0,b)$ algebra admits a non-degenerate bilinear and thus one can not have a Chern-Simons gauge theory formulation with them. However, they may appear in a three-dimensional gravity description, where we also need to have a spin 2 generator, that comes from the $(a=0,b=-1)$ sector. In the present work, we have demonstrated that the asymptotic symmetry algebra of a spin 3 gravity theory on flat spacetime has both the $W(0,-1)$ and $W(0,-2)$ algebras as subalgebras. We have also constructed a dual boundary field theory for this higher spin gravity theory by using the Chern-Simons/Wess-Zumino-Witten correspondence.