244 research outputs found
Constructions of free commutative integro-differential algebras
In this survey, we outline two recent constructions of free commutative
integro-differential algebras. They are based on the construction of free
commutative Rota-Baxter algebras by mixable shuffles. The first is by
evaluations. The second is by the method of Gr\"obner-Shirshov bases.Comment: arXiv admin note: substantial text overlap with arXiv:1302.004
A system of difference equations with elliptic coefficients and Bethe vectors
An elliptic analogue of the deformed Knizhnik-Zamolodchikov equations is
introduced. A solution is given in the form of a Jackson-type integral of Bethe
vectors of the XYZ-type spin chains.Comment: 20 pages, AMS-LaTeX ver.1.1 (amssymb), 15 figures in LaTeX picture
environment
Interaction of Reggeized Gluons in the Baxter-Sklyanin Representation
We investigate the Baxter equation for the Heisenberg spin model
corresponding to a generalized BFKL equation describing composite states of n
Reggeized gluons in the multi-color limit of QCD. The Sklyanin approach is used
to find an unitary transformation from the impact parameter representation to
the representation in which the wave function factorizes as a product of Baxter
functions and a pseudo-vacuum state. We show that the solution of the Baxter
equation is a meromorphic function with poles (lambda - i r)^{-(n-1)} (r= 0,
1,...) and that the intercept for the composite Reggeon states is expressed
through the behavior of the Baxter function around the pole at lambda = i . The
absence of pole singularities in the two complex dimensional lambda-plane for
the bilinear combination of holomorphic and anti-holomorphic Baxter functions
leads to the quantization of the integrals of motion because the holomorphic
energy should be the same for all independent Baxter functions.Comment: LaTex, 48 pages, 1 .ps figure, to appear in Phys. Rev.
Separation of variables for the quantum SL(2,R) spin chain
We construct representation of the Separated Variables (SoV) for the quantum
SL(2,R) Heisenberg closed spin chain and obtain the integral representation for
the eigenfunctions of the model. We calculate explicitly the Sklyanin measure
defining the scalar product in the SoV representation and demonstrate that the
language of Feynman diagrams is extremely useful in establishing various
properties of the model. The kernel of the unitary transformation to the SoV
representation is described by the same "pyramid diagram" as appeared before in
the SoV representation for the SL(2,C) spin magnet. We argue that this kernel
is given by the product of the Baxter Q-operators projected onto a special
reference state.Comment: 26 pages, Latex style, 9 figures. References corrected, minor
stylistic changes, version to be publishe
From finite geometry exact quantities to (elliptic) scattering amplitudes for spin chains: the 1/2-XYZ
Initially, we derive a nonlinear integral equation for the vacuum counting
function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus
paralleling similar results by Kl\"umper \cite{KLU}, achieved through a
different technique in the {\it antiferroelectric regime}. In terms of the
counting function we obtain the usual physical quantities, like the energy and
the transfer matrix (eigenvalues). Then, we introduce a double scaling limit
which appears to describe the sine-Gordon theory on cylindrical geometry, so
generalising famous results in the plane by Luther \cite{LUT} and Johnson et
al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to
excitations, we derive scattering amplitudes involving solitons/antisolitons
first, and bound states later. The latter case comes out as manifestly related
to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this
nonlinear integral equations framework was contrived to deal with finite
geometries, we prove it to be effective for discovering or rediscovering
S-matrices. As a particular example, we prove that this unique model furnishes
explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe}
and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description
of unknown integrable field theories.Comment: Article, 41 pages, Late
Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets
We consider correlation functions of the spin-\half XXX and XXZ Heisenberg
chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive
representations for various correlation functions in terms of determinants of
Fredholm integral operators.Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos correcte
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