Bergman and Calder\'on projectors for Dirac operators

TL;DR

This paper constructs and analyzes Calderón and Bergman projectors for Dirac operators on spin manifolds with boundary, using conformal covariance and scattering theory, and introduces conformally covariant powers of the Dirac operator.

Contribution

It provides a natural construction of Calderón and Bergman projectors for Dirac operators using scattering theory and conformal covariance, and develops conformally covariant Dirac powers.

Findings

Calderón projector expressed via scattering operators.

Schwartz kernels of projectors analyzed.

Conformally covariant Dirac powers constructed.

Abstract

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calder\'on projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g=\bar{g}/\rho^2$ where $\rho$ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$. We show that $({\rm Id}+\tilde{S}(0))/2$ is the orthogonal Calder\'on projector, where $\tilde{S}(\lambda)$ is the holomorphic family in $\{\Re(\lambda)\geq 0\}$ of normalized scattering operators constructed in our previous work, which are classical pseudo-differential of order $2\lambda$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any spin manifold.