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

TL;DR

This paper connects the Arnold conjecture with the Atiyah-Patodi-Singer index theorem by using spectral flow and localization to estimate the number of periodic trajectories in Hamiltonian systems.

Contribution

It introduces a novel approach combining spectral flow, index theory, and path integrals to provide a lower bound on periodic trajectories, supporting the Arnold conjecture.

Findings

Spectral flow computed via Atiyah-Patodi-Singer index theorem.

Path integral evaluated using localization methods.

Lower bound consistent with the Arnold conjecture.

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.