939 research outputs found
Monopoles and modifications of bundles over elliptic curves
Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle. Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the case R × (elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of theta-functions with characteristic
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
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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