Dirac spectral flow on contact three manifolds I: Eigensection estimates and spectral asymmetry

Let $Y$ be a compact, oriented $3$-manifold with a contact form $a$ and a metric $\mathrm{d}s^2$. Suppose that $F \to Y$ is a principal bundle with structure group $U(2) = SU(2) \times {}_{\lbrace \pm 1 \rbrace} S^1$ such that $F / S^1$ is the principal $SO(3)$ bundle of orthonormal frames for $TY$. A unitary connection $A_0$ on the Hermitian line bundle $F \times {}_{\mathrm{det} \: U(2)} \mathbb{C}$ determines a self-adjoint Dirac operator $\mathcal{D}_0$ on the $\mathbb{C}^2$-bundle $F \times {}_{U(2)} \mathbb{C}^2$.

The contact form a can be used to perturb the connection $A_0$ by $A_0 - {ira}$. This associates a one parameter family of Dirac operators $\mathcal{D}_r$ for $r \geq 0$. When $r \gg 1$, we establish a sharp sup-norm estimate on the eigensections of $\mathcal{D}_r$ with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from $\mathcal{D}_0$ to $\mathcal{D}_r$. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than $\mathcal{O} (r^{\frac{3}{2}})$. We also relate the $\eta$-invariant of $\mathcal{D}_r$ to certain spectral asymmetry function involving only the small eigenvalues of $\mathcal{D}_r$.

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Received 28 February 2015

Published 26 July 2017