Koszul complexes, Birkhoff normal form and the magnetic Dirac operator

Abstract

We consider the semiclassical Dirac operator coupled to a magnetic potential on a large class of manifolds, including all metric contact manifolds. We prove a sharp Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuations combined with the Birkhoff normal form and local index theory.