A Calder\'on Problem for the Dirac operator with chiral boundary conditions

TL;DR

This paper introduces a boundary conjugation map for the Dirac operator with chiral boundary conditions on spin manifolds, demonstrating it can determine the manifold's metric, connection, and spinor bundle from boundary data.

Contribution

It defines a new boundary map for the Dirac operator with chiral boundary conditions and proves it uniquely determines the manifold's geometric and gauge data in various dimensions.

Findings

Boundary conjugation map is a pseudodifferential operator of order 0.

The map's symbol determines the metric and connection modulo gauge.

Manifolds and bundles can be recovered from boundary data.

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.