## 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 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 SL_{2}(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 SL_{2}(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.

Original language | English |
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Pages (from-to) | 2685-2715 |

Number of pages | 31 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Issue number | 11 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Publisher Copyright:© European Mathematical Society 2018.

## Keywords

- Conformal blocks
- Phylogenetics
- Principal bundles

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics