The algebra of conformal blocks

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. In the SL${}_2(\mathbb{C})$ case, these degenerations result in toric varieties which appear in the theory of phylogenetic statistical varieties, and the study of integrable systems in the moduli of rank 2 vector bundles. We conclude with a combinatorial proof that the Cox ring of the moduli stack of quasi-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.