The algebra of conformal blocks

  • Christopher Manon

    George Mason University, Fairfax, USA

Abstract

For each simply connected, simple complex group we show that the direct sum of all vector bundles of conformal blocks on the moduli stack of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve agrees with the Cox ring of the moduli of quasi-parabolic principal -bundles on . We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. In the SL 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 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.

Cite this article

Christopher Manon, The algebra of conformal blocks. J. Eur. Math. Soc. 20 (2018), no. 11, pp. 2685–2715

DOI 10.4171/JEMS/822