\(\text{GR}=\text{SW}\): counting curves and connections.
type: Item
instance of: scholarly article
title: \(\text{GR}=\text{SW}\): counting curves and connections.
zbMATH Open document ID: 1040.53096
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
full work available at URL: 3 2.pdf
review text: The equivalence of Gromov-Witten and Seiberg-Witten invariants for symplectic manifolds is one of the most important results proved in geometry in the 1990's. The author announced the relationship in [Math. Res. Lett. 1, 809--822 (1994; Zbl 0853.57019)] and [Math. Res. Lett. 2, 221--238 (1995; Zbl 0854.57020)]. The detailed proof appears in three main parts. The first part is in [J. Am. Math. Soc. 9, 845--918 (1996; Zbl 0867.53025)], the second part is (*) [in J. Differ. Geom. 51, 203--334 (1999; Zbl 1036.53066)], and this paper contains the final part of the proof. The relevant version of the Gromov-Witten invariants was defined in [J. Differ. Geom. 44, 818--893 (1996; Zbl 0883.57020)]. Seiberg- Witten theory has the reputation of Gauge theory made simple. In general it is possible to produce short proofs of results in Seiberg-Witten gauge theory. The proof of the equivalence between Gromov-Witten invariants and Seiberg-Witten invariants, however, takes 300 pages of detailed analysis. We will give a rough statement of the equivalence and outline of the proof of the equivalence in this review. The precise statement may be found in the paper under review together with a more detailed outline of the proof. Of course, this paper also includes the complete proof of this important result. On a symplectic manifold the Seiberg-Witten invariant is a map, \(SW:H^2(X;\mathbb Z)\to\wedge^\ast H^1(X;\mathbb Z)\). A symplectic manifold has a canonical spin structure. Given any second cohomology class there is a corresponding line bundle that may be used to twist the canonical spin structure. The Seiberg-Witten equations depend on this structure. In local coordinates, they are a system of seven PDE's in eight unknown functions. There is a symmetry that effectively allows one to ignore one function to get an elliptic system. There is a \(\mu\)-map from the homology of \(X\) to the cohomology of the moduli space of solutions to the Seiberg-Witten equations modulo symmetry. The Seib…
reviewed by: David Auckly
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1782615
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: Seiberg-Witten invariants
zbMATH Keywords: symplectic manifolds
zbMATH Keywords: spin structure
author: Clifford Henry Taubes
Wikidata QID: Q115193650
MaRDI profile type: Publication
OpenAlex ID: W1491467571
DOI: 10.4310/JDG/1214425348
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description: scientific article; zbMATH DE number 1782615