On the Arnold conjecture and the Atiyah-Patodi-Singer index theorem

The Arnold conjecture yields a lower bound to the number of periodic classi cal trajectories in a Hamiltonian system. Here we count these trajectories with the help of a path integral, which we inspect using properties of the spectral flow of a Dirac operator in the background of a Sp(2N) valued gaug e field. We compute the spectral flow from the Atiyah-Patodi-Singer index t heorem, and apply the results to evaluate the path integral using localizat ion methods. In this manner we find a lower bound to the number of periodic classical trajectories which is consistent with the Arnold conjecture. (C) 1999 Published by Elsevier Science B.V. All rights reserved.