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 ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\tau$. At the level of classical mechanics the deformation parameter $\tau$ 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 $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ 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 $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ 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 $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ 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