ar X iv : h ep - t h / 93 12 12 3 v 2 7 J an 1 99 4 Affine Gelfand - Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem turns out to be equivalent to classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly. Introduction In the seventies M.Adler[A], I.M.Gelfand and L.A.Dickey [GD] discovered a natural Poisson structure on the space of n-th order differential operators on the circle with highest coefficient 1 which is now called the (second) Gelfand-Dickey bracket. This bracket arises in the theory of nonlinear integrable equations under various names (nKdV-structure, classical Wn-algebra). B.L.Feigin proposed to consider and study symplectic leaves for the Gelfand-Dickey bracket – a problem motivated by the fact that for n = 2 these symplectic leaves are orbits of coadjoint representation of the Virasoro algebra. A classification of symplectic leaves for the Gelfand-Dickey bracket and a description of their adjacency were given in [OK]. It turned out that locally symplectic leaves are labeled by one of the following: 1) conjugacy classes in the group GLn; 2) orbits of the coadjoint representation of the affine Lie algebra ĝln; 3) equivalence classes of flat vector bundles on the circle of rank n (these three things are in one-to-one correspondence). Moreover, adjacency of symplectic leaves is the same as that for conjugacy classes, orbits and vector bundles. Finally, the codimension of a symplectic leaf is equal to any of the following: 1) the dimension of the centralizer of the corresponding conjugacy class; 2) the codimension of the corresponding coadjoint orbit; 3) the dimension of the space of flat global sections of the bundle of endomorphisms of the corresponding flat vector bundle. In Section 1 of this paper we define an “affine” analogue of the Gelfand-Dickey bracket. It is realized on the space of n-th order differential operators on an elliptic curve which are polynomials in ∂ with smooth coefficients and highest coefficient 1. The reason to consider such brackets is a search for an appropriate two-dimensional counterpart of the theory of affine Lie algebras. One can show that the “affine” analogue of the Drinfeld-Sokolov reduction [DS] sends the linear Poisson bracket