Seiberg--Witten invariants and surface singularities

Abstract

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and `polygonal') singularities, and Brieskorn-Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister-Turaev sign refined torsion (or, equivalently, the Seiberg-Witten invariant) of a rational homology 3-manifold M, provided that M is given by a negative definite plumbing. These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel-Stern and Neumann-Wahl.