  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.
