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 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, p) E agrees with the Cox ring of the moduli of quasi-parabolic principal G-bundles on (C, p) E . We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. In the SL₂(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 principal SL₂(C)-bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees 2,3, and 4.