The algebra of $SL_3(\mathbb{C})$ conformal blocks

Abstract

We construct and study a family of toric degenerations of the algebra of conformal blocks for a stable marked curve $(C, \vec{p})$ with structure group $SL_3(\mathbb{C}).$ We find that this algebra is Gorenstein. 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. Along the way we recover polyhedral rules for counting conformal blocks originally due to Senechal, Mathieu, Kirillov, and Walton.