Bergman and Calderón projectors for Dirac operators

For a Dirac operator \(D_{\bar{g}}\) over a spin compact Riemannian manifold with boundary \((\bar{X},\bar{g})\), we give a new construction of the Calderon projector on \(\partial\bar{X}\) and of the associated Bergman projector on the space of L2 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 with the complete conformal metric \(g=\bar{g}/\rho^{2}\) where ρ is a smooth function on \(\bar{X}\) which equals the distance to the boundary near \(\partial\bar{X}\). We show that \(\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))\) is the orthogonal Calderon projector, where \(\tilde{S}(\lambda)\) is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.