33 research outputs found
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 matrices whose commutator differ from the identity by a
matrix of rank are used to construct bispectral differential operators with
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 , 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
of bispectral matrix differential operators is different than
those previously studied in that acts from the left, but from the
right on a common eigenmatrix.Comment: 16 page
Old and New Fields on Super Riemann Surfaces
The ``new fields" or ``superconformal functions" on 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 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 be a complex Lie group and denote the group of maps from
the unit circle into , of a suitable class. A differentiable
map from a manifold into , is said to be of \emph{connection
order } if the Fourier expansion in the loop parameter of the
-family of Maurer-Cartan forms for , namely F_\lambda^{-1}
\dd F_\lambda, is of the form . 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 map,
into a pair of simpler maps of order and respectively.
Conversely, one could construct such a harmonic map from any pair of
and 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 map, for ,
splits uniquely into a pair of and 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