Gaussian beams and Calder\'on problems for connections at large frequency

We consider the inverse problem of recovering a connection on a complex vector bundle over a compact smooth Riemannian manifold with boundary from a Dirichlet-to-Neumann (DN) map at a high fixed frequency. We construct Gaussian beams using the language of jet bundles and show that their value at the boundary can be recovered from those DN maps. This allows us to show injectivity up to gauge on manifolds whose non-abelian X-ray transform is injective. We also study DN maps with a cubic nonlinearity and show how to recover the broken non-abelian X-ray transform from them. This transform maps a connection to its parallel transport along broken geodesics with endpoints on the boundary. We show that the broken non-abelian X-ray transform is always injective up to gauge equivalence, regardless of the manifold's geometry.