Superpseudodifferential Operators Research Articles

A new super KP system and a characterization of the Jacobians of arbitrary algebraic super curves

A set of super-commuting vector fields is defined on the super Grassmannians. A characterization of the Jacobian varieties of super curves (super Schottky problem) is established in the following manner: Every finite dimensional integral manifold of these vector fields has a canonical structure of the Jacobian variety of an algebraic super curve, and conversely, the Jacobian variety of an arbitrary algebraic super curve is obtained in this way. The vector fields restricted on the super Grassmannian of index 0|0 give a completely integrable system of partial super differential equations which gives a new supersymmetric generalization of the KP system. Thus every finite-dimensional solution of this new system gives rise to a Jacobian variety of an algebraic super curve. The correspondence between this super Grassmannian and the group of monic super pseudo-differential operators of order zero (the super Sato correspondence) is also established. Table of

Superanalysis: Theory of generalized functions and pseudodifferential operators

A theory of distributions on an infinite-dimensional superspace is constructed. Feynman and Gaussian supermeasures are defined on the basis of this theory and superpseudodifferential operators are introduced. We obtain the basic formulas in the theory of superpseudodifferential operators and the Feynman-Kac formula for the symbol of the evolution operator (here, in contrast to the algebraic approach, the functional integral is an integral over a space of actual trajectories in the phase superspace). The differential calculus in graded /Lambda/ modules proposed in this paper is needed for the introduction of infinite-dimensional superpseudodifferential operators and also for nonsequential definition of the Feynman integral.