202 research outputs found
Integrable Systems and Factorization Problems
The present lectures were prepared for the Faro International Summer School
on Factorization and Integrable Systems in September 2000. They were intended
for participants with the background in Analysis and Operator Theory but
without special knowledge of Geometry and Lie Groups. In order to make the main
ideas reasonably clear, I tried to use only matrix algebras such as
and its natural subalgebras; Lie groups used are either GL(n)
and its subgroups, or loop groups consisting of matrix-valued functions on the
circle (possibly admitting an extension to parts of the Riemann sphere). I hope
this makes the environment sufficiently easy to live in for an analyst. The
main goal is to explain how the factorization problems (typically, the matrix
Riemann problem) generate the entire small world of Integrable Systems along
with the geometry of the phase space, Hamiltonian structure, Lax
representations, integrals of motion and explicit solutions. The key tool will
be the \emph{% classical r-matrix} (an object whose other guise is the
well-known Hilbert transform). I do not give technical details, unless they may
be exposed in a few lines; on the other hand, all motivations are given in full
scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems,
Classical r-matrices and Factorization problem
Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. II. General Semisimple Case
The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov
reduction procedure to q-difference operators associated with arbitrary
semisimple Lie algebras. This leads to a new elliptic deformation of the Lie
bialgebra structure on the associated loop algebra. The related classical
r-matrix is explicitly described in terms of the Coxeter transformation. We
also present a cross-section theorem for q-gauge transformations which
generalizes a theorem due to R.Steinberg.Comment: 19 pp., AMS-LaTeX. The paper replaces a temporarily withdrawn text;
the first part (written by E. Frenkel, N. Reshetikhin, and M. A.
Semenov-Tian-Shansky) is available as q-alg/970401
Classical and Quantum Nonultralocal Systems on the Lattice
We classify nonultralocal Poisson brackets for 1-dimensional lattice systems
and describe the corresponding regularizations of the Poisson bracket relations
for the monodromy matrix . A nonultralocal quantum algebras on the lattices for
these systems are constructed.For some class of such algebras an
ultralocalization procedure is proposed.The technique of the modified
Bethe-Anzatz for these algebras is developed.This technique is applied to the
nonlinear sigma model problem.Comment: 33 pp. Latex. The file is resubmitted since it was spoiled during
transmissio
Dual parametrization of GPDs versus the double distribution Ansatz
We establish a link between the dual parametrization of GPDs and a popular
parametrization based on the double distribution Ansatz, which is in prevalent
use in phenomenological applications. We compute several first forward-like
functions that express the double distribution Ansatz for GPDs in the framework
of the dual parametrization and show that these forward-like functions make the
dominant contribution into the GPD quintessence function. We also argue that
the forward-like functions with contribute to the
leading singular small- behavior of the imaginary part of DVCS
amplitude. This makes the small- behavior of \im A^{DVCS} independent
of the asymptotic behavior of PDFs. Assuming analyticity of Mellin moments of
GPDs in the Mellin space we are able to fix the value of the -form factor in
terms of the GPD quintessence function and the forward-like function
.Comment: 18 pages, 5 figures. A version that appeared in Eur. Phys. J. A. Some
of the statements were refined and misprints in the formulas were correcte
Dual parametrization of generalized parton distributions in two equivalent representations
The dual parametrization and the Mellin-Barnes integral approach represent
two frameworks for handling the double partial wave expansion of generalized
parton distributions (GPDs) in the conformal partial waves and in the
-channel partial waves. Within the dual parametrization
framework, GPDs are represented as integral convolutions of forward-like
functions whose Mellin moments generate the conformal moments of GPDs. The
Mellin-Barnes integral approach is based on the analytic continuation of the
GPD conformal moments to the complex values of the conformal spin. GPDs are
then represented as the Mellin-Barnes-type integrals in the complex conformal
spin plane. In this paper we explicitly show the equivalence of these two
independently developed GPD representations. Furthermore, we clarify the
notions of the fixed pole and the -form factor. We also provide some
insight into GPD modeling and map the phenomenologically successful
Kumeri\v{c}ki-M\"uller GPD model to the dual parametrization framework by
presenting the set of the corresponding forward-like functions. We also build
up the reparametrization procedure allowing to recast the double distribution
representation of GPDs in the Mellin-Barnes integral framework and present the
explicit formula for mapping double distributions into the space of double
partial wave amplitudes with complex conformal spin.Comment: 56 pages, 3 figure
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