  Dirac fermions in graphene can be subjected to non-abelian gauge fields by
implementing certain modulations of the carbon site potentials. Artificial
graphene, engineered with a lattice of CO molecules on top of the surface of
Cu, offers an ideal arena to study their effects. In this work, we show by
symmetry arguments how the underlying CO lattice must be deformed to obtain
these gauge fields, and estimate their strength. We also discuss the
fundamental differences between abelian and non-abelian gauge fields from the
Dirac electrons point of view, and show how a constant (non-abelian) magnetic
field gives rise to either a Landau level spectrum or a quadratic band
touching, depending on the gauge field that realizes it (a known feature of
non-abelian gauge fields known as the Wu-Yang ambiguity). We finally present
the characteristic signatures of these effects in the site-resolved density of
states that can be directly measured in the current molecular graphene
experiment, and discuss prospects to realize the interaction induced broken
symmetry states of a quadratic touching in this system.
