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On the Arnold conjecture and the Atiyah-Patodi-Singer index theorem. (English) Zbl 1058.53510

Summary: 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.

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
58J20 Index theory and related fixed-point theorems on manifolds
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