Lattice gauge theories and the Florentino conjecture

We investigate the relations between the space of classes of $\operatorname{SU}(2)$-representations of the fundamental group of a Riemann surface $\Sigma_\Gamma$ equipped with a trinion decomposition corresponding to a 3-valent graph $\Gamma$ and the $\operatorname{SU}(2)$ theory on $\Gamma$. We construct a section of the standard map of the orbit space of the gauge theory on $\Sigma_\Gamma$ onto that of the gauge theory on $\Gamma$. As an application, we prove a conjecture of Florentino.