Bergman and Calderón projectors for Dirac operators

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 Calderón 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 $g=\bar{g}/ρ^2$ where $ρ$ 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 Calderón projector, where $\tilde{S}(λ)$ is the holomorphic family in $\{\Re(λ)\geq 0\}$ of normalized scattering operators constructed in our previous work, which are classical pseudo-differential of order $2λ$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any spin manifold.