Hamiltonian symplectomorphisms and the Berry phase

Abstract

On the space $\mathcal{L}$, of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed $\mathbf{Z}$-valued 1-form $\Omega$. If $\Omega$ vanishes, the prequantization map can be extended to a group representation. On $\mathcal{L}$ one can define an action integral as an $\mathbf{R}/\mathbf{Z}$-valued function, and the cohomology class $[\Omega ]$ is the obstruction to the lifting of that action integral to an $\mathbf{R}$-valued function. The form $\Omega$ also defines a natural grading on ${\pi }_1(\mathcal{L})$.

18 pages