The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the electromagnetic field.
What happens physically is that after parallel transport along a loop, the wave function acquires a phase difference, physically invisible, but that can lead to observable interference.
Now, a "phase" in a $U(1)$ gauge theory can generalize in two different ways if we go to non-abelian gauge theories:
1) It could remain a phase, leading to no physical difference (except interference);
2) It could become a more general $G$ transformation, where $G$ is the gauge group (like $SU(2)$ or $SU(3)$), and for example change the color of a quark after a loop.
What does quantum field theory predict? What would happen if we set a strong (or weak) equivalent of the Aharonov-Bohm effect (despite the obvious experimental difficulties)?
If 1) is true: wouldn't it mean that the meaningful (curved) part of the holonomy is in fact Abelian? If 2) is true: wouldn't it violate (for quarks) conservation of color charge?
Any reference would be good, as long as it gets to the point.