The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.

ABSTRACT. We note that the Berezin integral, which is ill-defined for noncompact supermanifolds, is a distribution with support on the underlying manifold. This leads to the discovery of correction terms in the Berezinian transformation law and thereby eliminates the boundary ambiguities.1. Introduction. A perspective on the Berezin integral. A real (p, q)-dimensional COO supermanifold is a ringed space (M, d)

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This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order r > 1. As a result, the bispectrality of a large family of algebras of rank r is demonstrated. An involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters, generalizing Wilson's rank one bispectral involution. Spectral geometry and the relationship to the Sato grassmannian are discussed.

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