Compactifications of character varieties and skein relations on conformal blocks

Abstract

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 $M_{C_{\Gamma}}(G)$ associated to a singular stable curve $C_{\Gamma}$ 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 $M_{C_{\Gamma}}(SL_2(\mathbb{C}))$ is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves $(C_{\Gamma}, \vec{p}_{\Gamma})$.