Higher Spin Chern-Simons Theory and the Super Boussinesq hierarchy

TL;DR

This paper establishes a correspondence between supersymmetric higher spin gravity in Chern-Simons theory and the N=2 super Boussinesq hierarchy, revealing their shared dynamics and Poisson structures.

Contribution

It constructs a detailed map linking supersymmetric Chern-Simons higher spin gravity with the super Boussinesq hierarchy, highlighting their equivalent evolution equations and Hamiltonian structures.

Findings

Time evolution equations of both theories coincide.

Poisson structure of Chern-Simons theory matches super Boussinesq hierarchy.

Provides a new integrable systems perspective on higher spin gravity.

Abstract

In this paper we construct a map between a solution of supersymmetric Chern-Simons higher spin gravity based on the superalgebra $sl(3|2)$ with Lifshitz scaling and the $N=2$ super Boussinesq hierarchy. We show that under this map the time evolution equations of both theories coincide. In addition, we identify the Poisson structure of the Chern-Simons theory induced by gauge transformation with the second Hamiltonian structure of the super Boussinesq hierarchy.