The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere

TL;DR

This paper explores the quaternionic KP hierarchy's connection to conformally immersed 2-tori in the 4-sphere, analyzing spectral curves and their role in classifying finite spectral genus tori.

Contribution

It extends the KP hierarchy framework to quaternions, compares different spectral curve notions, and links them to the classification of conformally immersed 2-tori in S^4.

Findings

The quaternionic KP hierarchy is equivalent to the Davey-Stewartson II hierarchy.

Three notions of spectral curves are compared and related.

The classification of finite spectral genus tori is based on a singular spectral curve.

Abstract

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of quaternionic holomorphic 2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4). After describing how the Sato-Segal-Wilson construction of KP solutions (particularly solutions of finite type) carries over to this quaternionic setting, we compare three different notions of "spectral curve": the QKP spectral curve, which arises from an algebra of commuting differential operators; the (unnormalised) Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The latter two are shown to be images of the QKP spectral curve, which need not be smooth. Moreover, it is a singularisation of this QKP spectral curve, rather than the normalised Floquet multiplier curve, which determines the classification of conformally immersed 2-tori of finite spectral genus.