The geometry of the super KP flows

A supersymmetric generalization of the Krichever map is used to construct algebro-geometric solutions to the various super Kadomtsev-Petviashvili (SKP) hierarchies. The geometric data required consist of a suitable algebraic supercurve of genusg (generallynot a super Riemann surface) with a distinguished point and local coordinates (z, ) there, and a generic line bundle of degreeg–1 with a local trivialization near the point. The resulting solutions to the Manin-Radul SKP system describe coupled deformations of the line bundle and the supercurve itself, in contrast to the ordinary KP system which deforms line bundles but not curves. Two new SKP systems are introduced: an integrable Jacobian system whose solutions describe genuine Jacobian flows, deforming the bundle but not the curve; and a nonintegrable maximal system describing independent deformations of bundle and curve. The Kac-van de Leur SKP system describes the same deformations as the maximal system, but in a different parametrization.