Seiberg-Witten invariants and surface singularities

By A. N'emethi, Liviu I. Nicolaescu. 2001

103 citations recorded by the source corpus.

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 verications are based on a result which describes (in terms of the plumbing graph) the Reidemeister{Turaev sign rened torsion (or, equivalently, the Seiberg{ Witten invariant) of a rational homology 3{manifold M , provided that M is given by a negative denite plumbing. These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel{Stern and Neumann{Wahl.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/245654df784a6865306e9e1255bcd2ae97304b49

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Citations in the corpus

Links and analytic invariants of superisolated singularities (2003, background): “A G ] 1 6 O ct 2 00”
The Seiberg-Witten invariants of negative definite plumbed 3-manifolds (2010, methodology): “i based on the surgery formulas of [MW02]). In terms of Γ, a combinatorial formula for the Casson–Walker invariant can be deduced from Lescop’s book [L96], while the Turaev’s torsion is determined in [NN02]. Nevertheless, this expression of the torsion is based on a Dedekind–Fourier sum, which, in m”
On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds (2003, methodology): “d by the Heegaard Floer theory) provide a (conjecturally optimal) topological upper bound for h1(L), where L is a holomorphic line bundle on a resolution of (X,0). For the extension of the conjecture [12] to arbitrary spinc structures, see [11]. For diﬀerent deﬁnitions regarding surface singularitie”
The geometric genus of splice-quotient singularities (2006, unknown): “There are many studies on this invariant related to the above issue; for example, [1], [4], [13], [16], [7], [8].”
Complex surface singularities with integral homology sphere links (2003, background): “Though the Casson Invariant Conjecture for this case follows, it had already been proven in [26] (by a much less conceptual proof), and more recently by Collin and Saveliev [5] using equivariant Casson invariants and by Némethi and Nicolaescu [23] in a more general context.”
Fivebranes and 3-manifold homology (2016, background): “This agrees with the known result for torsion of negative definite plumbed 3-manifolds [65]:”
Lattice cohomology of normal surface singularities (2007, methodology): “Then (∗) is true by [25], Corollary1.5. Moreover, by (4.4.1), eu(H∗) = −(−1)+3−1 = 3, (K2 +s)/8 = −1. On the other hand, the Casson invariant of M is −2 (using, e.g., the formula of Ra¸tiu, see (5.3) [16]). Hence, (∗∗) is valid as well. 5.2.6. Remarks. (a) The above identities are not valid (in this”