On the intersection theory of moduli spaces of parabolic bundles

This paper concerns the intersection numbers of tautological classes on moduli spaces of parabolic bundles on a smooth projective curve. We show that such intersection numbers are completely determined by wall-crossing formulas, Hecke isomorphisms, and flag bundle structures and resulting Weyl symmetry. As applications of these ideas, we prove the Newstead--Earl--Kirwan vanishing -- and a natural strengthening in terms of Chern filtrations -- and the Virasoro constraints for parabolic bundles. Both of these results were already known for moduli of stable bundles without parabolic structure, but even in those cases our proofs are new and independent of the existing ones. We use a Joyce style vertex algebra formulation of wall-crossing, and define intersection numbers even in the presence of strictly semistable parabolic bundles; all of our results hold in that setting as well.