Generalized intersection pairings on moduli spaces of vector bundles over a curve

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.