The Calder\'on problem for connections

TL;DR

This paper extends the Calderón problem for connections on vector bundles to more general manifolds, establishing uniqueness results for the connection from boundary measurements and introducing new methods for constructing solutions.

Contribution

It generalizes previous results to cases with less restrictive geometric conditions and develops new techniques for constructing CGO solutions on vector bundles.

Findings

Uniqueness of connection recovery from Dirichlet-to-Neumann map.

Construction of Gaussian beam and CGO solutions for vector bundles.

Reduction to a non-abelian X-ray transform problem.

Abstract

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of the transversal manifold being simple -- we generalise this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of \cite{CTA}. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.