Conformal Blocks in type C at level one

TL;DR

This paper explores the properties of conformal blocks vector bundles for symplectic Lie algebras at level one on moduli spaces, revealing conditions under which their divisors align with those of $sl_2$ and establishing the polyhedral nature of the generated cone.

Contribution

It establishes a precise relationship between conformal blocks divisors for $sp_{2 ext{ exttwosuperior}} ext{ exttwosuperior}$ at level one and $sl_2$ at level $ ext{ exttwosuperior}$, including conditions for their equivalence and the structure of the divisor cone.

Findings

First Chern classes match conformal blocks divisors for $sl_2$ when vector bundles have rank one or zero.

All such divisors become equal for large enough $ ext{ exttwosuperior}$.

The cone generated by these divisors is polyhedral.

Abstract

We investigate the behavior of vector bundles of conformal blocks for $sp_{2\ell}$ at level one on $\bar{M}_{0,n}$. We show their first Chern classes are equivalent to conformal blocks divisors for $sl_2$ at level $\ell$ if and only if the corresponding vector bundles have rank one or zero, and all become equal when the Lie algebra $\ell$ is large enough. As a consequence of these results, we conclude the cone generated by these divisors is polyhedral.