W Gravity From Chern--Simons Theory

TL;DR

This paper derives a covariant action from 3D Chern--Simons theory with $Sl(N,R)$ gauge group, exhibiting $W_N$ symmetry and connecting it to Toda theory and anomalies.

Contribution

It introduces a new covariant action invariant under $W_N$ transformations derived from Chern--Simons theory, linking it to Toda theory and anomaly interpretation.

Findings

Derived $S_{cov}$ invariant under $W_N$ transformations

Connected $S_{cov}$ with Toda theory

Provided anomaly interpretation of the action

Abstract

Starting with three dimensional Chern--Simons theory with gauge group $Sl(N,R)$, we derive an action $S_{cov}$ invariant under both left and right $W_N$ transformations. We give an interpretation of $S_{cov}$ in terms of anomalies, and discuss its relation with Toda theory.