A Calderón Problem for the Dirac operator with chiral boundary conditions
Carlos Valero
TL;DR
This work develops a Calderón-type inverse problem for the Dirac operator with chiral boundary conditions on a spin manifold with boundary, introducing the boundary conjugation map $\Theta_{g,A,m}$ as a first-order boundary data operator. The authors show $\Theta_{g,A,m}$ is a pseudodifferential operator of order 0 whose principal symbol encodes the boundary metric up to conformal equivalence, and whose lower-order symbols recover the normal derivatives of the metric and unitary connection, thereby determining the boundary Taylor series of $(g,A)$ under suitable nondegeneracy conditions (notably $n\ge3$, $m\neq0$, and $m^2$ not in the Dirichlet spectrum of $D_A^2$). In the real-analytic setting, they prove recovery of the entire manifold, twisted spinor bundle, and connection from $\Theta_{g,A,m,\mathscr{U}}$, up to a flat line bundle obstruction $L$ that encodes possible spin-structure twists; with extra hypotheses this obstruction is trivial and global gauge equivalence follows. For $n=2$, analogous boundary-determination and reconstruction results hold without the auxiliary bundle in the stated regime. The approach hinges on relating $\Theta$ to the Dirichlet-to-Neumann map for $D_A^2-m^2$, performing detailed symbol calculus, and leveraging Green's kernels to realize a global analytic reconstruction. The results extend the Calderón program to first-order Dirac operators and contribute a robust framework for recovering geometric and gauge data from boundary spinor information.
Abstract
We consider on a spin manifold with boundary a Dirac operator $D_A$ with chiral boundary conditions, twisted by a unitary connection $A$. When $m$ is not in the chiral spectrum of $D_A$, we define an analogue of the Dirichlet-to-Neumann map for the Dirac equation $D_A - m$, which we call the boundary conjugation map, and show that it is a pseudodifferential operator of order $0$ on the boundary. We show that in dimension greater than 2, its symbol determines the Taylor series of the metric and connection modulo gauge on the boundary when $m \neq 0$ and $m^2$ is not in the Dirichlet spectrum of $D_A^2$. We go on to show that a real-analytic Riemannian manifold and twisted spinor bundle with twisted spin connection can be recovered from its boundary conjugation map. Under further hypotheses, one can recover the unitary connection up to global gauge equivalence and the complex spinor bundles. Similar results hold in dimension $2$ when the auxiliary bundle and connection are absent.