\(\text{Gr}\Rightarrow\text{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions.
type: Item
instance of: scholarly article
title: \(\text{Gr}\Rightarrow\text{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions.
zbMATH Open document ID: 1036.53066
author: Clifford Henry Taubes
published in: Journal of Differential Geometry
publication date: 2000-06-26T00:00:00Z
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. Cliff Taubes announced the relationship in [Math. Res. Lett. 1, No. 6, 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, No. 3, 845--918 (1996; Zbl 0867.53025); see also the revised verson with a corrected Section 6e in Proc. First IP Lecture Series, Vol. 2, Internat. Press, to appear]. This present paper contains the second part of the argument, and the third part is contained in [J. Differ. Geom. 52, No. 3, 453--609 (1999; Zbl 1040.53096)]. 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 theory. The proof of the equivalence between Gromov-Witten invariants and Seiberg-Witten invariants, however, takes 300 pages of detailed analysis. The main result in this paper establishes the existence of a solution to suitably perturbal Seiberg-Witten equations arising from a pseudo-holomorphic curve in a symplectic manifold. After the introduction, Taubes reviews the Seiberg-Witten equations, the geometry of pseudo-holomorphic curves, and extensively reviews the vortex equations which Taubes worked on early in his career. The vortex equations are important in this context, because they arise as the dimensional reduction of the Seiberg-Witten equations to two real dimensions. In the second section Taubes constructs a bundle over a pseudo-holomorphic curve with fiber isomorphic to the space of solutions to the vortex equations over \(\mathbb C\). The key point in this section is the construction of an approximate solution to the perturbed Seiberg-Witten equations out of a section of the vortex bundle. The Seiberg-Witten equations state that the twisted Dirac operator of a spinner field is triv…
reviewed by: David Auckly
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1465065
zbMATH Keywords: knot and link surgery
zbMATH Keywords: 4-manifolds
zbMATH Keywords: cylindrical ends
zbMATH Keywords: gluing
zbMATH Keywords: Seiberg-Witten theory
Wikidata QID: Q115194205
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214425068
OpenAlex ID: W1481574008
DOI: 10.4310/JDG/1214425068
description: scientific article; zbMATH DE number 1465065