Let $\mathcal{M}_{g,h,r,s}$ be a Riemann surface with genus $g$, $h$ boundary components, r interior marked points, and $s$ marked points on the boundary $\partial \mathcal{M} = \Sigma$. In the case where $s=0$, the isomorphism classes of Principal G-bundles over $\mathcal{M}_{g,h,r}$ are in bijection to $\dfrac{Hom( \pi_1(\mathcal{M}_{g,h,r}), G)}{G}$ where those homomorphisms have to have match the conjugacy classes specified on the boundaries. My question is how this generalizes to the case $s>0$. Do we need to specify conjugacy classes on each segment (i.e. in between the marked points) of the marked boundaries? Is there a bijection to something resembling $\dfrac{Hom( \pi_1(\mathcal{M}_{g,h,r}), G)}{G}$? I guess the simplest context could be for a principal bundle over a disk with 2 marked points on its boundary. I would greatly appreciate any good references that explain the construction of principal bundles over $\mathcal{M}_{g,h,r,s}$.
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