Paper detail

Compactifications of character varieties and skein relations on conformal blocks

Let $M_C(G)$ be the moduli space of semistable principal $G-$bundles over a smooth curve $C$. We show that a flat degeneration of this space $M_{C_Γ}(G)$ associated to a singular stable curve $C_Γ$ contains the free group character variety $\mathcal{X}(F_g, G)$ as a dense, open subset, where $g = genus(C).$ In the case $G = SL_2(\mathbb{C})$ we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of $M_{C_Γ}(SL_2(\mathbb{C}))$ is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves $(C_Γ, \vec{p}_Γ)$.