Connections of Berry and Hannay type for moving Lagrangian submanifolds

Abstract

Berry's phase is the holonomy of the natural connection on the canonical circle bundle over projectivized quantum Hilbert space. Given a symplectic manifold P, a classical limit of this phase is constructed as the holonomy of a Berry connection over a finite-codimensional isodrastic foliation (defined by constancy of action integrals) on the space of lagrangian submanifolds in P equipped with smooth densities of total measure 1. If the densities are determined by a Kähler metric compatible with the symplectic structure, the curvature of the Berry connection at the lagrangian submanifold L is given by a simple formula involving curvature of L. In particular, the Berry connection defines a homotopy invariant for isodrastic loops of minimal lagrangian submanifolds in a simply connected Kähler manifold. A similar invariant constructed by the author for loops of symplectomorphisms is also a special case of the classical Berry phase. A classical analogue of Berry's phase was discovered by Hannay for moving families of completely integrable systems. Following Berry and Hannay, we interpret Hannay's angles as the holonomy of a Hannay connection on a bundle of tori over the isodrastic foliation on the space of lagrangian toral layers consisting of lagrangian tori with a flat affine structure and an extension of this structure to the first infinitesimal neighborhood. Finally, we show that the Hannay angles are the derivatives with respect to action variables of the classical Berry phase.