36,568 research outputs found
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
Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version
The aim of this paper is two-fold. First, we define symplectic maps between
Hitchin systems related to holomorphic bundles of different degrees. We call
these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs
bundles. They are constructed by means of the Hecke correspondence of the
underlying holomorphic bundles. SHC allows to construct B\"{a}cklund
transformations in the Hitchin systems defined over Riemann curves with marked
points. We apply the general scheme to the elliptic Calogero-Moser (CM) system
and construct SHC to an integrable \SLN Euler-Arnold top (the elliptic
\SLN-rotator). Next, we propose a generalization of the Hitchin approach to
2d integrable theories related to the Higgs bundles of infinite rank. The main
example is an integrable two-dimensional version of the two-body elliptic CM
system. The previous construction allows to define SHC between the
two-dimensional elliptic CM system and the Landau-Lifshitz equation.Comment: 39 pages, the definition of the symplectic Hecke correspondence is
explained in details, typos corrected, references adde
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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