  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.
