Dirac spectral flow on contact three manifolds I: eigensection estimates and spectral asymmetry
Resource
Dirac spectral flow on contact three manifolds I: eigensection estimates and spectral asymmetry
Journal
Journal of Symplectic Geometry
Pages
541-602
Date Issued
2013-07-17
Date
2013-07-17
Author(s)
Abstract
Let Y be a compact, oriented 3-manifold with a contact form a and a metric ds2. Suppose that F → Y is a principal bundle with structure group U(2) = SU(2) ×{±1} S1 such that F/S1 is the principal SO(3) bundle of orthonormal frames for TY. A unitary connection A0 on the Hermitian line bundle F ×det U(2) C determines a self-adjoint Dirac operator D0 on the C2-bundle F ×U(2) C2. The contact form a can be used to perturb the connection A0 by A0 − ira. This associates a one parameter family of Dirac operators Dr for r ≥ 0. When r >> 1, we establish a sharp sup-norm estimate on the eigensections of Dr with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from D0 to Dr. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than O(r3/4). We also relate the η-invariant of Dr to certain spectral asymmetry function involving only the small eigenvalues of Dr.
Type
journal article
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