On the Arnold Conjecture and the Atiyah-Patodi-Singer Index Theorem

Abstract

The Arnold conjecture yields a lower bound to the number of periodic classical 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 gauge field. We compute the spectral flow from the Atiyah-Patodi-Singer index theorem, and apply the results to evaluate the path integral using localization methods. In this manner we find a lower bound to the number of periodic classical trajectories which is consistent with the Arnold conjecture.