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

Abstract

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $d{s}^2$. Suppose that $F\to Y$ is a principal bundle with structure group $U(2)=SU(2){\times }_{\pm 1}{S}^1$ 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 $r\ge 0$. 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 $D_r$.

arXiv comment48 pages. All comments welcome