Calder\'on problem for Yang-Mills connections

TL;DR

This paper addresses the inverse problem of determining Yang-Mills connections on vector bundles from boundary measurements, establishing uniqueness results and analyzing the Dirichlet-to-Neumann map's properties.

Contribution

It proves uniqueness of Yang-Mills connections from boundary data in certain cases and characterizes the DN map as an elliptic pseudodifferential operator.

Findings

Uniqueness of trivial line bundle connections in smooth category

Unique determination of bundle and connection with multiple measurements

DN map is an elliptic pseudodifferential operator of order one

Abstract

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection, but at the cost of having more measurements. Also, we prove that the DN map is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.