Hitchin systems, higher Gaudin operators and $r$-matrices

Abstract

Hitchin systems, higher Gaudin operators and r-matrices B. Enriquez and V. Rubtsov Abstract. We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of CP 1 and SL n -bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical r-matrix for Hitchin systems for a punctured elliptic curve, and GL nbundles, and (for n = 2) the corresponding quantum system. Introduction. In [13], N. Hitchin introduced a class of integrable systems, attached to a complex curve X and a semisimple Lie group G. The problem of quantization of these systems was then connected by A. Beilinson and V. Drinfel'd to the Langlands program. This quantization makes use of differential operators on the moduli space of G-bundles on X, obtained from the action of the center of the local completion of the enveloping algebra of a Kac-Moody algebra, at the critical level.This program can also be carried out in the case of curves with marked points. In the particular case of the punctured CP 1 , the action of the center of the enveloping algebra was studied by B. Feigin, E. Frenkel and N. Reshetikhin in [6]; they obtained an integrable system whose first operators are identical to Gaudin's operators ([9]).In this paper, we consider the question of expressing the action of higher central elements. We first note, that the Adler-Kostant-Symes (AKS) scheme, which allows to write families of commuting operators ([2], [14], [21]), can be applied in the present situation, and then show that the higher Hamiltonians obtained in [6], coincide with those. So our problem turns out to be equivalent to expressing higher central elements in the enveloping algebras at critical level, a problem which was solved by several authors ([10], [12]). Here we show how the results of [10] can be used to derive a simple expression of higher Gaudin Hamiltonians.We then turn to the case of punctured elliptic curves. We show that the integrability of Hitchin's system can be deduced from an r-matrix argument. Here r-matrix relations contain additional terms, due to an invariance under the Cartan algebra action. The r-matrix depends on the moduli parameters, so it reminds dynamical r-matrices. In the case of one puncture, our L-operator and r-matrix seem connected with those considered respectively by I. Krichever and A. Gorsky and N. Nekrasov in [15] and [11], and H. Braden, T. Suzuki and E. Sklyanin [5], [19]. It is also analogous to the r-matrix appearing in the work of G. Felder and C. Wieczerkowski on the Knizhnik-Zamolodchikov-Bernard equation on elliptic curves ([7]). We give the form of the first Hamiltonians in this case; one of them contains a Weierstass potential, and so is analogous to the Calogero-Moser Hamiltonian. We comput...

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