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

TL;DR

This paper investigates the spectral flow and eigensection estimates of Dirac operators on contact 3-manifolds, establishing sharp bounds and analyzing spectral asymmetry as the perturbation parameter grows large.

Contribution

It provides sharp sup-norm estimates for eigensections of Dirac operators on contact 3-manifolds and analyzes the asymptotic behavior of spectral flow and eta-invariants for large perturbations.

Findings

Sharp sup-norm estimate for eigensections with small eigenvalues

Subleading order term of spectral flow is smaller than r^{3/2}

Relation between eta-invariant and spectral asymmetry of small eigenvalues

Abstract

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $ds^2$. Suppose that $F\to Y$ is a principal bundle with structure group $U(2) = SU(2)\times_{\pm1}S^1$ 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 $F\times_{\det U(2)}\mathbb{C}$ determines a self-adjoint Dirac operator $D_0$ 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 $D_r$ for $r\geq0$. When $r>>1$, we establish a sharp sup-norm estimate on the eigensections of $D_r$ with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from $D_0$ to $D_r$. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than the order of $r^{\frac{3}{2}}$. We also relate the $\eta$-invariant of $D_r$ to certain spectral asymmetry function involving only the small eigenvalues of $D_r$.