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

TL;DR

This paper explores the algebraic structure of conformal blocks for $SL_3(C)$, constructing toric degenerations, establishing Gorenstein property, and deriving bounds on generators and relations, with implications for algebraic and geometric understanding.

Contribution

It introduces a family of toric degenerations for the conformal blocks algebra and determines its Gorenstein property, providing new bounds and polyhedral counting rules.

Findings

The algebra of conformal blocks is Gorenstein.

Bounds on degrees of generators and relations are established.

Polyhedral rules for counting conformal blocks are recovered.

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.