Constraining all possible Korteweg-de Vries type hierarchies

Abstract

The Lie algebra of symmetries generated by the left-moving current $j={\partial }_{-}\varphi$ 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.

v1, 23 pages, 1 ancillary file; v2, 26 pages, additional comparisons with literature; v3, 27 pages, published version

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