\(\text{GR}=\text{SW}\): counting curves and connections.
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 Seiberg-Witten invariant of a cohomology class applied to a collection of homology cycles is the integral \[ \int_{\mathcal M} \nu(\gamma_1)\wedge \nu(\gamma_2)\dots \wedge \nu(\gamma_p)\wedge \nu(pt)^{d-p/2}. \] The Gromov-Witten invariant is also a map \(Gr:H^2(X:\mathbb Z)\to \wedge^\ast H^1(X:\mathbb Z)\). Roughly, \(Gr(e)(\gamma_1\dots\gamma_p)\) is equal to the number of pseudo-holomorphic submanifolds of \(X\) that are dual to \(e\) and meet each curve \(\gamma_R\) in exactly one point. A submanifold is called pseudo-holomorphic if each tangent space is complex with respect to a suitable almost complex structure. Four of the functions in the local description of the Seiberg-Witten equations combine into the coefficients of a unitary connection on the line bundle associated to the spin structure. The other four functions combine into a spinner field. For symplectic manifolds, the positive spin bundle may be identified with \(E\oplus(K^{-1}\otimes E)\). Let \(\alpha\) be the component of the spinner field in \(E\). The main result of [J. Am. Math. Soc. 9, No. 3, 845--918 (1996; Zbl 0867.53025)] stated that \(\alpha^{-1}(0)\) is approximated by a pseudo-holomorphic submanifold when the component of the perturbation in the direction of the symplectic form is sufficiently large. It is well known that integrals of closed forms may be interpreted as intersection numbers for smooth manifolds. The first main step in this proof is an intersection theory reinterpretation of the Seiberg-Witten invariant. This goes as follows: Let \(\Omega\) denote a finite subset of \(X\), and \(\Gamma\) denote a collection of curves in \(X\). The relevant intersection is given as the set \({\mathcal M}^{(r)}_{\Gamma, \Omega}\), consisting of solutions to the Seiberg-Witten equations with \(\alpha^{-1}(0)\cap\Omega\not=\emptyset\) and \(\alpha^{-1}(0)\cap\gamma\not=\emptyset\) for some \(\gamma\) in \(\Gamma\). When this is regular, the Seiberg-Witten invariant is just a signed count of these points. More generally, the Seiberg-Witten invariant is obtained as a count of these points with multiplicities. Taubes proceeds to prove the equivalence for a sequence of cases of increasing complexity throughout the paper until he attains the final result. The first cases are when \({\mathcal M}^{(r)}_{\Gamma,\Omega}\) consists of one or zero regular points (corresponding to \(E\) trivial and part of the blow-up formula). One of the key tools used is a special set of sections of a vortex bundle, \(\mathbb Z_0\). The vortex bundle is defined over any pseudo-holomorphic submanifold and has fiber isomorphic to the space of vortices of fixed degree over \(\mathbb C\). The special set of sections is cut out by a non-linear elliptic equation. In the previous paper, [see (*), loc. cit.], the Gromov-Witten invariant was related to the points in \(\mathbb Z_0\) and these points were shown to be embedded in \({\mathcal M}^{(r)}_{\Gamma,\Omega}\). The new technical results in this paper amount to showing that this embedding is surjective and showing that the multiplicities of points in each model agree. The case when all parts of \(\mathbb Z_0\) are regular is covered before the general case. The final issue addressed is the count of multiply covered tori. This is accomplished by proving that the multiplicities are universal numbers and then evaluating these numbers on \(T^4\) with the help of a wall-crossing formula.
scientific article; zbMATH DE number 1782615