Integrable systems and the boundary dynamics of higher spin gravity on AdS$_3$

TL;DR

This paper establishes a connection between higher spin gravity in AdS3 and integrable hierarchies, specifically the modified Boussinesq hierarchy, by introducing new boundary conditions and analyzing the boundary dynamics.

Contribution

It introduces novel boundary conditions for higher spin gravity that reveal an integrable structure at the boundary, linking bulk geometry to the modified Boussinesq hierarchy.

Findings

Boundary conditions lead to integrable boundary dynamics.

The phase space and conserved charges match the modified Boussinesq hierarchy.

The geometric Miura map is derived from the bulk theory.

Abstract

We introduce a new set of boundary conditions for three-dimensional higher spin gravity with gauge group $SL(3,\mathbb{R})\times SL(3,\mathbb{R})$, where its dynamics at the boundary is described by the members of the modified Boussinesq integrable hierarchy. In the asymptotic region the gauge fields are written in the diagonal gauge, where the excitations go along the generators of the Cartan subalgebra of $sl(3,\mathbb{R})\oplus sl(3,\mathbb{R})$. We show that the entire integrable structure of the modified Boussinesq hierarchy, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are obtained from the asymptotic structure of the higher spin theory. Furthermore, its known relation with the Boussinesq hierarchy is inherited from our analysis once the asymptotic conditions are re-expressed in the highest weight gauge. Hence, the Miura map is recovered from a purely geometric construction in the bulk. Black holes that fit within our boundary conditions, the Hamiltonian reduction at the boundary, and the generalization to higher spin gravity with gauge group $SL(N,\mathbb{R})\times SL(N,\mathbb{R})$ are also discussed.