Parallel Transport along Seifert Manifolds and Fractional Monodromy


In this work, together with Nikolay Martynchuk, we investigated fractional monodromy in two-degree of freedom integrable Hamiltonian systems. In such systems, Seifert manifolds naturally appear by taking closed paths in the image of the integral map and considering the preimage of such paths in the phase space. Fractional monodromy is then defined through the parallel transport of homology cycles along Seifert manifolds, expanding on an idea from Uncovering fractional monodromy. It turns out that fractional monodromy is determined by the exceptional fibers of the Seifert manifold and by the fixed points of the circle action. This result, in the same spirit as results for monodromy of torus bundles, provides a deep understanding of fractional monodromy and its origins. This is the first time that fractional monodromy has been associated to a certain type of singularities in integrable Hamiltonian systems (fixed points of a circle action). Details can be found in our paper which is now on the arXiv.