\(\text{Gr}\Rightarrow\text{SW}\): from pseudo-holomorphic curves to Seiberg-Witten solutions.
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 trivial and the self-dual part of the curvature of the connection is a quadratic function of the spinner field plus a perturbation. On a symplectic manifold Taubes showed that the best perturbation to use is the sum of the self-dual part of the curvature of a reference connection plus a multiple of the symplectic form. Dilations on \(\mathbb C\) act on solutions to the vortex equations. The dilation parameter corresponds to the coefficient in front of the symplectic form. The approximate solutions get closer to honest solutions as this parameter increases. However, not every section will produce a Seiberg-Witten solution in the limit. This is the topic of the third section of this paper. In this technical section Taubes defines a subset, \({\mathcal Z}_0\), of sections of the vortex bundle that do converge to Seiberg-Witten solutions in the large dilation limits. This subset is described as the solutions to a non-linear system of differential equations. Intuitively, these special sections may be thought of as \(J\text{-holomorphic}\) sections of the vortex bundle. The remainder of this third section is devoted to the structure of this special set of sections. The fourth section of this paper introduces a proposition stating the existence of a map from a subset of the special sections to the Seiberg-Witten moduli space together with the key estimates for this map. This proposition is proved in sections four and five via the contraction mapping principal. This requires the large number of detailed estimates that are contained in these two sections. The most general version of this map is given in proposition 5.2. The sixth and final section of this paper continues the study of the map described in proposition 5.2. In particular, the map is shown to be an embedding and a Kuranishi structure for the perturbed Seiberg-Witten moduli space is given.
scientific article; zbMATH DE number 1465065