Hamiltonian symplectomorphisms and the Berry phase

On the space L, of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed Z-valued 1-form Omega . If Omega vanishes, the prequantization map can be extended to a group rep resentation. On L one can define an action integral as an R/Z-valued functi on, and the cohomology class [Omega] is the obstruction to the lifting of t hat action integral to an R-valued function. The form Omega also defines a natural grading on pi (1)(L). (C) 2001 Elsevier Science B.V. All rights res erved.