Abstract
Conformal block divisors in type A on M0;n are shown to satisfy new symmetries when levels and ranks are interchanged in non-standard ways. A connection with the quantum cohomology of Grassmannians reveals that these divisors vanish above the critical level. '0: the critical level c(slr+1; ~ ) and the theta level (g;~ ). The critical level, which we introduce, is related to an interpretation of the ranks of the bundles with g = slr+1 in terms of the quantum cohomology of the Grassmannian. If ' is greater than either bound, then D g; ~ = 0 (Theorem 1.3 and Remark 1.5). While equal for r = 1, the critical and theta levels are generally distinct, reecting dierent aspects of the weights ~ . As an application of vanishing, we give sucient conditions for divisors D g;~ to be extremal in the nef cone (Propositions 5.3 and 5.4), and show that the morphisms they dene factor
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