Koszul complexes, Birkhoff normal form and the magnetic Dirac operator

TL;DR

This paper develops a semi-classical analysis of the magnetic Dirac operator on various manifolds, establishing a local Weyl law and eta invariant bounds using advanced microlocal techniques.

Contribution

It introduces a novel approach combining almost analytic continuations, Birkhoff normal form, and local index theory to analyze the magnetic Dirac operator without a Fourier integral parametrix.

Findings

Proved a sharp local Weyl law for the magnetic Dirac operator.

Established bounds on the eta invariant in the semi-classical setting.

Extended analysis to a broad class of manifolds including metric contact manifolds.

Abstract

We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local 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.