Stable pair invariants of surfaces and Seiberg-Witten invariants

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The moduli space of stable pairs on a local surface X=Kₛ is in general non-compact. The action of C* on the fibres of X induces an action on the moduli space and the stable pair invariants of X are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section S ⊂ X. Sometimes there are no other contributions, e.g. when the curve class β is irreducible. We relate these surface stable pair invariants to the Poincaré invariants of Dürr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of S by work of Dürr-Kabanov-Okonek and Chang-Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for X = Kₛ implies Taubes' GW/SW correspondence for S. (2) When pg(S) = 0, the difference of surface stable pair invariants in class β and Kₛ - β is a universal topological expression.