The Algebra of $$ {{\mathbf{SL}}_3}\left( \mathbb{C} \right) $$ Conformal Blocks

Abstract

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL3( $$ \mathbb{C} $$ ) bundles on a smooth, marked curve (C, $$ \vec{p} $$ ): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $$ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $$ of quasi-parabolic principal SL3( $$ \mathbb{C} $$ ) bundles on (C, $$ \vec{p} $$ ). Along the way we recover positive polyhedral rules for counting conformal blocks.