## 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

Funding Information:Acknowledgments. We thank Weronika Buczyńska, Edward Frenkel, Noah Giansiracusa, Kaie Kub-jas, Shrawan Kumar, Eduard Looijenga, Diane Maclagan, John Millson, Steven Sam, Bernd Sturm-fels, David Swinarski, and Filippo Viviani for useful discussions. We also thank the reviewer for many helpful suggestions. This paper was mostly written at the fall 2009 Introductory Workshop in Tropical Geometry at MSRI. This work was supported by the NSF fellowship DMS-0902710.

Publisher Copyright:

© European Mathematical Society 2018.

## Keywords

- Conformal blocks
- Phylogenetics
- Principal bundles

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics