If I want to have a space of $G$-connections on a Riemann surface, I can take the fundamental group on the surface, represent its generators on $G$ and take (up to conjugation) those representation maps (which would be the same thing as holonomies along appropriate loops with respect to $G$-connection) as generators on the space of connections.
Now, how does it work if I would like to add a spin structure on the surface, and replace group $G$ with a $\Bbb Z_2$-graded group?