266 research outputs found
A few remarks on integral representation for zonal spherical functions on the symmetric space
The integral representation on the orthogonal groups for zonal spherical
functions on the symmetric space is used to obtain a
generating function for such functions. For the case N=3 the three-dimensional
integral representation reduces to a one-dimensional one.Comment: Latex file, 10 pages, amssymb.sty require
Double coset construction of moduli space of holomorphic bundles and Hitchin systems
We present a description of the moduli space of holomorphic vector bundles
over Riemann curves as a double coset space which is differ from the standard
loop group construction. Our approach is based on equivalent definitions of
holomorphic bundles, based on the transition maps or on the first order
differential operators. Using this approach we present two independent
derivations of the Hitchin integrable systems. We define a "superfree" upstairs
systems from which Hitchin systems are obtained by three step hamiltonian
reductions. A special attention is being given on the Schottky parameterization
of curves.Comment: 19 pages, Late
Classical integrable systems and soliton equations related to eleven-vertex R-matrix
In our recent paper we suggested a natural construction of the classical
relativistic integrable tops in terms of the quantum -matrices. Here we
study the simplest case -- the 11-vertex -matrix and related
rational models. The corresponding top is equivalent to the 2-body
Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on
its description. We give different descriptions of the integrable tops and use
them as building blocks for construction of more complicated integrable systems
such as Gaudin models and classical spin chains (periodic and with boundaries).
The known relation between the top and CM (or RS) models allows to re-write the
Gaudin models (or the spin chains) in the canonical variables. Then they assume
the form of -particle integrable systems with constants. We also
describe the generalization of the top to 1+1 field theories. It allows us to
get the Landau-Lifshitz type equation. The latter can be treated as non-trivial
deformation of the classical continuous Heisenberg model. In a similar way the
deformation of the principal chiral model is also described.Comment: 24 page
Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
We construct special rational Knizhnik-Zamolodchikov-Bernard
(KZB) equations with punctures by deformation of the corresponding
quantum rational -matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is . At the level of classical mechanics the deformation parameter
allows to extend the previously obtained modified Gaudin models to the
modified Schlesinger systems. Next, we notice that the identities underlying
generic (elliptic) KZB equations follow from some additional relations for the
properly normalized -matrices. The relations are noncommutative analogues of
identities for (scalar) elliptic functions. The simplest one is the unitarity
condition. The quadratic (in matrices) relations are generated by
noncommutative Fay identities. In particular, one can derive the quantum
Yang-Baxter equations from the Fay identities. The cubic relations provide
identities for the KZB equations as well as quadratic relations for the
classical -matrices which can be halves of the classical Yang-Baxter
equation. At last we discuss the -matrix valued linear problems which
provide Calogero-Moser (CM) models and Painleve equations
via the above mentioned identities. The role of the spectral parameter plays
the Planck constant of the quantum -matrix. When the quantum
-matrix is scalar () the linear problem reproduces the Krichever's
ansatz for the Lax matrices with spectral parameter for the CM models. The linear problems for the quantum CM models generalize the KZ
equations in the same way as the Lax pairs with spectral parameter generalize
those without it.Comment: 26 pages, minor correction
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