The Algebra of Conformal Blocks

TL;DR

This paper demonstrates that conformal blocks form a flat sheaf of algebras over moduli stacks, with degenerations to toric varieties linked to biological phylogenetic models, and explores the algebraic structure of Cox rings for principal bundles.

Contribution

It establishes a new algebraic framework for conformal blocks as a flat sheaf of algebras and connects degenerations to toric varieties and biological models.

Findings

Conformal blocks form a flat sheaf of commutative algebras.

Degenerations of these algebras are toric for SL_2(C).

Cox rings are generated by low-level conformal blocks with bounded relations.

Abstract

For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve $(C, \vec{p})$ agrees with the Cox ring of the moduli of quasi-parabolic principal $G-$bundles on $(C, \vec{p})$. We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. These degenerations are toric in the case $G = SL_2(\mathbb{C}),$ and the resulting toric varieties are shown to be isomorphic to phylogenetic algebraic varieties from mathematical biology. We conclude with a proof that the Cox ring of the moduli stack of qausi-parabolic $SL_2(\mathbb{C})$ principal bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees $2, 3,$ and 4.