102 research outputs found
Separation of Variables in the Classical Integrable SL(3) Magnetic Chain
There are two fundamental problems studied by the theory of hamiltonian
integrable systems: integration of equations of motion, and construction of
action-angle variables. The third problem, however, should be added to the
list: separation of variables. Though much simpler than two others, it has
important relations to the quantum integrability. Separation of variables is
constructed for the magnetic chain --- an example of integrable model
associated to a nonhyperelliptic algebraic curve.Comment: 13 page
Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
We construct the Baxter Q-operator and the representation of the Separated
Variables (SoV) for the homogeneous open SL(2,R) spin chain. Applying the
diagrammatical approach, we calculate Sklyanin's integration measure in the
separated variables and obtain the solution to the spectral problem for the
model in terms of the eigenvalues of the Q-operator. We show that the
transition kernel to the SoV representation is factorized into the product of
certain operators each depending on a single separated variable. As a
consequence, it has a universal pyramid-like form that has been already
observed for various quantum integrable models such as periodic Toda chain,
closed SL(2,R) and SL(2,C) spin chains.Comment: 29 pages, 9 figures, Latex styl
Vertex operator approach to semi-infinite spin chain : recent progress
Vertex operator approach is a powerful method to study exactly solvable
models. We review recent progress of vertex operator approach to semi-infinite
spin chain. (1) The first progress is a generalization of boundary condition.
We study spin chain with a triangular boundary, which
gives a generalization of diagonal boundary [Baseilhac and Belliard 2013,
Baseilhac and Kojima 2014]. We give a bosonization of the boundary vacuum
state. As an application, we derive a summation formulae of boundary
magnetization. (2) The second progress is a generalization of hidden symmetry.
We study supersymmetry spin chain with a diagonal
boundary [Kojima 2013]. By now we have studied spin chain with a boundary,
associated with symmetry , and
[Furutsu-Kojima 2000, Yang-Zhang 2001, Kojima 2011,
Miwa-Weston 1997, Kojima 2011], where bosonizations of vertex operators are
realized by "monomial" . However the vertex operator for
is realized by "sum", a bosonization of boundary
vacuum state is realized by "monomial".Comment: Proceedings of 10-th Lie Theory and its Applications in Physics,
LaTEX, 10 page
Noncompact SL(2,R) spin chain
We consider the integrable spin chain model - the noncompact SL(2,R) spin
magnet. The spin operators are realized as the generators of the unitary
principal series representation of the SL(2,R) group. In an explicit form, we
construct R-matrix, the Baxter Q-operator and the transition kernel to the
representation of the Separated Variables (SoV). The expressions for the energy
and quasimomentum of the eigenstates in terms of the Baxter Q-operator are
derived. The analytic properties of the eigenvalues of the Baxter operator as a
function of the spectral parameter are established. Applying the diagrammatic
approach, we calculate Sklyanin's integration measure in the separated
variables and obtain the solution to the spectral problem for the model in
terms of the eigenvalues of the Q-operator. We show that the transition kernel
to the SoV representation is factorized into a product of certain operators
each depending on a single separated variable.Comment: 29 pages, 12 figure
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
New Formula for the Eigenvectors of the Gaudin Model in the sl(3) Case
We propose new formulas for eigenvectors of the Gaudin model in the \sl(3)
case. The central point of the construction is the explicit form of some
operator P, which is used for derivation of eigenvalues given by the formula , where , fulfil
the standard well-know Bethe Ansatz equations
High Energy QCD: Stringy Picture from Hidden Integrability
We discuss the stringy properties of high-energy QCD using its hidden
integrability in the Regge limit and on the light-cone. It is shown that
multi-colour QCD in the Regge limit belongs to the same universality class as
superconformal =2 SUSY YM with at the strong coupling
orbifold point. The analogy with integrable structure governing the low energy
sector of =2 SUSY gauge theories is used to develop the brane picture
for the Regge limit. In this picture the scattering process is described by a
single M2 brane wrapped around the spectral curve of the integrable spin chain
and unifying hadrons and reggeized gluons involved in the process. New
quasiclassical quantization conditions for the complex higher integrals of
motion are suggested which are consistent with the duality of the
multi-reggeon spectrum. The derivation of the anomalous dimensions of the
lowest twist operators is formulated in terms of the Riemann surfacesComment: 37 pages, 3 figure
Bethe Ansatz and boundary energy of the open spin-1/2 XXZ chain
We review recent results on the Bethe Ansatz solutions for the eigenvalues of
the transfer matrix of an integrable open XXZ quantum spin chain using
functional relations which the transfer matrix obeys at roots of unity. First,
we consider a case where at most two of the boundary parameters
{{,,,}} are nonzero. A generalization of
the Baxter equation that involves more than one independent is
described. We use this solution to compute the boundary energy of the chain in
the thermodynamic limit. We conclude the paper with a review of some results
for the general integrable boundary terms, where all six boundary parameters
are arbitrary.Comment: 6 pages, Latex; contribution to the XVth International Colloquium on
Integrable Systems and Quantum Symmetries, Prague, June 2006. To appear in
Czechoslovak Journal of Physics (2006); (v2) Typos corrected and a new line
added in the Acknowledgments sectio
An integrable discretization of the rational su(2) Gaudin model and related systems
The first part of the present paper is devoted to a systematic construction
of continuous-time finite-dimensional integrable systems arising from the
rational su(2) Gaudin model through certain contraction procedures. In the
second part, we derive an explicit integrable Poisson map discretizing a
particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the
contraction procedures enable us to construct explicit integrable
discretizations of the continuous systems derived in the first part of the
paper.Comment: 26 pages, 5 figure
A Symplectic Structure for String Theory on Integrable Backgrounds
We define regularised Poisson brackets for the monodromy matrix of classical
string theory on R x S^3. The ambiguities associated with Non-Ultra Locality
are resolved using the symmetrisation prescription of Maillet. The resulting
brackets lead to an infinite tower of Poisson-commuting conserved charges as
expected in an integrable system. The brackets are also used to obtain the
correct symplectic structure on the moduli space of finite-gap solutions and to
define the corresponding action-angle variables. The canonically-normalised
action variables are the filling fractions associated with each cut in the
finite-gap construction. Our results are relevant for the leading-order
semiclassical quantisation of string theory on AdS_5 x S^5 and lead to
integer-valued filling fractions in this context.Comment: 41 pages, 2 figures; added references, corrected typos, improved
discussion of Hamiltonian constraint
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