Asymptotic spectral flow for Dirac operators of disjoint Dehn twists

Abstract

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$. For any Dirac operator $\mathcal{D}$, we study the asymptotic behavior of the spectral flow between $\mathcal{D}$ and $\mathcal{D} + \mathrm{cl}(-\frac{ir}{2}a)$ as $r \to \infty$. If $a$ is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is $\mathcal{O}(r)$.