CMC hierarchy II: Non-commuting symmetries and affine Kac-Moody algebra

TL;DR

This paper extends the CMC hierarchy by incorporating non-commuting symmetries from the Virasoro algebra, utilizing affine Kac-Moody algebra and dressing transformations to define tau functions.

Contribution

It introduces a novel extension of the CMC hierarchy with non-commuting symmetries and provides explicit formulas for the tau function using affine Kac-Moody algebra techniques.

Findings

Defined a wave function for the extended hierarchy.

Derived a closed formula for the tau function.

Established a connection between spectral Killing fields and affine Kac-Moody algebra.

Abstract

Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additional formal Killing fields, and the corresponding spectral Killing field is defined by a purely algebraic formula up to an integrable extension. The extended CMC hierarchy is obtained by packaging these data into the associated affine Kac-Moody algebra valued Killing fields. The log of tau function of the extended CMC hierarchy is defined as the central component of the affine extension of the spectral Killing field. We give a closed formula for the tau function in terms of the determinant of the spectral Killing field.