Asymptotic spectral flow for Dirac operators of disjoint Dehn twists. (English) Zbl 1308.58014

After setting some conventions of Dirac operators on a closed oriented contact 3-manifold and spectral flow functions, a review on open book decompositions (OBD) and the constructions of Thurston-Winkelnkemper contact forms are provided. The Dirac equation on different regions of the OBD is given. On the region with trivial monodromy, almost zero modes of the Dirac operator are constructed by the Riemann-Roch theorem. The Dirac equation on the model manifold \(S^1 x S^2\) is studied. By combining the above results, a lower bound of the spectral flow function is obtained. Then Dirac equation is studied on the region where the monodromy is trivial. The upper bound of the spectral flow function is proved. The main result here states that if the monodromy of an OBD is the product of Dehn twists along disjoint circles, and \(a\) is the associated Thurston-Winkelnkemper contact form, then, with a certain adapted metric, the next order term of the spectral flow function of the canonical Dirac operator is of order \(O(r)\).


58J30 Spectral flows
53D35 Global theory of symplectic and contact manifolds
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