In the second of a pair of papers, we complete our geometric construction of “Lagrangian matching invariants” for smooth four-manifolds equipped with broken fibrations. We prove an index formula, a vanishing theorem for connected sums and an analogue of the Meng–Taubes formula. These results lend support to the conjecture that the invariants coincide with Seiberg–Witten invariants of the underlying four-manifold, and are in particular independent of the broken fibration.
"Lagrangian matching invariants for fibred four-manifolds: II." Geom. Topol. 12 (3) 1461 - 1542, 2008. https://doi.org/10.2140/gt.2008.12.1461