The Algebra of SL₃(ℂ) Conformal Blocks

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL₃(ℂ) bundles on a smooth, marked curve (C, 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 M_C,p→(SL₃(ℂ)) of quasi-parabolic principal SL₃(ℂ) bundles on (C, p→). Along the way we recover positive polyhedral rules for counting conformal blocks.