D-modules on 1|1 Supercurves

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with connection are studied, and examples are given where X is a superelliptic curve.Comment: 18 page

Canonical Noether symmetries and commutativity properties for gauge systems

For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.Comment: 22 pages; some references updated, an uncited reference deleted, minor style change

Tzitz\'eica transformation is a dressing action

We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitz\'eica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitz\'eica transformations, and discuss the group structure for these transformations

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case $r=1$, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs $(L, \Lambda)$ of bispectral matrix differential operators is different than those previously studied in that $L$ acts from the left, but $\Lambda$ from the right on a common $r\times r$ eigenmatrix.Comment: 16 page

Old and New Fields on Super Riemann Surfaces

The ``new fields" or ``superconformal functions" on $N=1$ super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated $N=2$ super Riemann surface. We confirm that, as originally defined, they do not form a super vector space.Comment: 9 pages, LaTex. Published version: minor changes for clarity, two new reference

Dressing Technique for Intermediate Hierarchies

A generalized AKNS systems introduced and discussed recently in \cite{dGHM} are considered. It was shown that the dressing technique both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies.Comment: 16 pages, LaTeX Report/no: DFTUZ/94/2

Generalized DPW method and an application to isometric immersions of space forms

Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely F_\lambda^{-1} \dd F_\lambda, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order $(_{-1}^1)$ map, into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively. Conversely, one could construct such a harmonic map from any pair of $(_{-1}^{-1})$ and $(_1^1)$ maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an application, we show that constant non-zero curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.Comment: Some typographical correction

Applications of Temperley-Lieb algebras to Lorentz lattice gases

Motived by the study of motion in a random environment we introduce and investigate a variant of the Temperley-Lieb algebra. This algebra is very rich, providing us three classes of solutions of the Yang-Baxter equation. This allows us to establish a theoretical framework to study the diffusive behaviour of a Lorentz Lattice gas. Exact results for the geometrical scaling behaviour of closed paths are also presented.Comment: 10 pages, latex file, one figure(by request