  We show that the dipole-dipole interaction between two Rydberg atoms can lead
to substantial Abelian and non-Abelian gauge fields acting on the relative
motion of the two atoms. We demonstrate how the gauge fields can be evaluated
by numerical techniques. In the case of adiabatic motion in a single internal
state, we show that the gauge fields give rise to a magnetic field that results
in a Zeeman splitting of the rotational states. In particular, the ground state
of a molecular potential well is given by the first excited rotational state.
We find that our system realises a synthetic spin-orbit coupling where the
relative atomic motion couples to two internal two-atom states. The associated
gauge fields are non-Abelian.
