Higher Spin Lifshitz Theories and the KdV-Hierarchy

TL;DR

This paper establishes a universal connection between three-dimensional higher spin Lifshitz theories and the KdV hierarchy, revealing that their conserved charges are independent of the Lifshitz scaling exponent.

Contribution

It demonstrates an explicit, z-independent map linking higher spin Lifshitz Chern-Simons theories to the KdV hierarchy using the Drinfeld-Sokolov formalism.

Findings

Existence of a universal map for all z and N

Conserved charges are independent of the Lifshitz scaling exponent

Connection established via the Drinfeld-Sokolov formalism

Abstract

In this paper three dimensional higher spin theories in the Chern-Simons formulation with gauge algebra $sl(N,R)$ are investigated which have Lifshitz symmetry with scaling exponent $z$. We show that an explicit map exists for all $z$ and $N$ relating the Lifshitz Chern-Simons theory to the $(n,m)$ element of the KdV hierarchy. Furthermore we show that the map and hence the conserved charges are independent of $z$. We derive these result from the Drinfeld-Sokolov formalism of integrable systems.