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We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones.

We study a certain family of Schrodinger operators whose eigenfunctions phi(x, lambda) satisfy a differential equation in the spectral parameter lambda of the form B(lambda, partial derivative-lambda)phi = THETA(x)phi. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class of bispectral potentials. This extends and complements a result of Duistermaat and Grunbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy

We recall some deformation theory of Susy-curves and study obstructions to projectedness of supermoduli spaces, both from a general standpoint and by means of the local "coordinate charts" most commonly used in the physical literature. We prove that these give rise to a projected atlas for genus g=2 only. © 1990 Springer-Verlag.

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations. © 1985 Springer-Verlag.

The occurrence of non-abelian anomalies in gauge theories and gravitation, first discovered via perturbative techniques, is now completely explained from the mathematical point of view by means of the family index theorem of Atiyah and Singer. Here we make contact between this approach and BRS cohomology, by showing that they yield the same non-abelian anomalies, provided a certain restriction to "local" functionals is not introduced from the very beginning. In particular, this solves the "unicity" problem for this kind of anomalies. Local BRS cohomology is still relevant for the abelian case. © 1985 Springer-Verlag.