Constraining all possible Korteweg-de Vries type hierarchies

TL;DR

This paper explores the structure of integrable deformations of 2D scalar conformal field theories, deriving constraints on Korteweg-de Vries type equations and confirming the infinite dimensionality of certain subalgebras.

Contribution

It characterizes the properties of mutually commuting subalgebras in the symmetry algebra, providing new constraints on KdV-type integrable equations.

Findings

Mutually commuting subalgebras are infinite dimensional.

Derived general properties for generators of such subalgebras.

Confirmed the infinite dimensionality of specific recently discovered subalgebras.

Abstract

The Lie algebra of symmetries generated by the left-moving current $j=\partial_-\phi$ in the $2d$ single scalar conformal field theory is infinite dimensional, exhibiting mutually commuting subalgebras. The infinite dimensional mutually commuting subalgebras define integrable deformations of the $2d$ single scalar conformal field theory which preserve the Poisson bracket structure. We study these mutually commuting subalgebras, finding general properties that the generators of such a subalgebra must satisfy. Along the way, we derive constraints on integrable equations of the Korteweg-de Vries type. We also confirm that the recently found $[j]=0,-1,-2$ mutually commuting subalgebras are infinite dimensional.