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 $G=S{L}_2(\mathbb{C})$ we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of $M_{C_{\Gamma }}(S{L}_2(\mathbb{C}))$ is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves $(C_{\Gamma },{\vec{p}}_{\Gamma })$.

36 page, 24 figures, final version

References in corpus (4)

• Toric degenerations of spherical varieties
• Conformal blocks revisited
• Toric geometry of $S{L}_2(\mathbb{C})$ free group character varieties from outer space
• Conformal blocks, Berenstein-Zelevinsky triangles and group-based models

Cited by in corpus (3)

• Global geometry on moduli of local systems for surfaces with boundary
• Wonderful Compactification of Character Varieties
• Newton-Okounkov polyhedra for character varieties and configuration spaces