  We show that for a simple surface with boundary the attenuated ray transform
in the presence of a unitary connection and a skew-Hermitian Higgs field is
injective modulo the natural obstruction for functions and vector fields. We
also show that the connection and the Higgs field are uniquely determined by
the scattering relation modulo a gauge transformation. The proofs involve a
Pestov type energy identity for connections together with holomorphic gauge
transformations which arrange the curvature of the connection to have definite
sign.
