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

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.