Area Preserving Diffeomorphisms and $W_{\infty}$ Symmetry in a $2+1$ Chern-Simons Theory

TL;DR

This paper explores the $W_{ olinebreak\infty}$ symmetry in a $2+1$ dimensional Chern-Simons gauge theory, revealing its role as canonical transformations and drawing analogies with string theory states.

Contribution

It demonstrates the action of $W_{\infty}$ generators as canonical transformations and connects discrete states in string theory with Landau level states in gauge theory.

Findings

$W_{\infty}$ symmetry acts as canonical transformations on the ground state.

Analogies established between string theory discrete states and Landau levels.

Insights into symmetry structures in $2+1$ gauge theories with Chern-Simons term.

Abstract

We discuss the $W_{\infty}$ symmetry in the $2+1$ gauge theory with the Chern-Simons term. It is shown that the generators of this symmetry act on the ground state as the canonical transformations in the phase space. We shall also discuss the analogy between discrete states in $c=1$ string theory and Landau level states in $2+1$ gauge theory with Chern-Simons term.