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arXiv · hep-th/9908138

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

Mauri Miettinen·Antti J. Niemi
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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.

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Mauri Miettinen, Antti J. Niemi. 1999-08-25. On the Arnold Conjecture and the Atiyah-Patodi-Singer Index Theorem. https://doi.org/10.1016/s0370-2693(99)00820-5

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