The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtainedby replacing the complex numbers with the quaternions, mutatis mutandis, in thestandard construction of the KP hierarchy equations and solutions; it isequivalent to what is often called the Davey-Stewartson II hierarchy. Thisarticle studies its relationship with the theory of quaternionic holomorphic2-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 quaternionicsetting, we compare three different notions of "spectral curve": the QKPspectral curve, which arises from an algebra of commuting differentialoperators; the (unnormalised) Floquet multiplier spectral curve for the relatedDirac operator; and the curve parameterising Darboux transforms of a conformal2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The lattertwo 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 thenormalised Floquet multiplier curve, which determines the classification ofconformally immersed 2-tori of finite spectral genus.
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