On an Equivalence of Divisors on (M)over-bar(0,n) from Gromov-Witten Theory and Conformal Blocks

We consider a conjecture that identifies two types of base point free divisors on M ¯ ,n. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on M ¯ ,n to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on M ¯ ,n for a large class, and we give sufficient conditions for the non-vanishing of these divisors.