The Algebra of Conformal Blocks

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.