The Seiberg-Witten invariants and symplectic forms

Abstract

Recently, Seiberg and Witten (see [SW1], [SW2], [W]) introduced a remarkable new equation which gives differential-topological invariants for a compact, oriented 4-manifold with a distinguished integral cohomology class. A brief mathematical description of these new invariants is given in the recent preprint [KM].My purpose here is to prove the following theorem:Main Theorem. Let X be a compact, oriented, 4 dimensional manifold with b 2+ ≥ 2. Let ω be a symplectic form on X with ω ∧ ω giving the orientation. Then the first Chern class of the associated almost complex structure on X has Seiberg-Witten invariant equal to ±1.(Note: There are no symplectic forms on X unless b 2+ and the first Betti number of X have opposite parity.)In a subsequent article with joint authors, a vanishing theorem will be proved for the Seiberg-Witten invariants of a manifold X, as in the theorem, which can be split by an embedded 3-sphere as X − ∪ X + where neither X − nor X + have negative definite intersection forms. Thus, no such manifold admits a symplectic form. That is, Corollary.Connect sums of 4-manifolds with non-negative definite intersection forms do not admit symplectic forms which are compatible with the given orientation. For example, when n > 1 and m ≥ 0, then ( #n CP 2 )#( #m CP 2 ) has no symplectic form which defines the given orientation.The Main Theorem also implies that the Seiberg-Witten invariant for the canonical class of a complex surface with b 2+ ≥ 3 is equal to ±1. However, this result is easy to prove directly, as there is just one nondegenerate solution in this case.

From Seiberg-Witten invariants to topological Green-Schwarz string

“…As we have showed in the previous section, the Seiberg-Witten theory (8), in the limit |ω| → ∞, localizes on the pseudo-holomorphic curves (14). Here we would like to concentrate on the physical string theory which is directly related to the previously studied field theory.…”

“…Among them there is the topological field theory behind the above N=2 Green-Schwarz string and its comparison with the theory of pseudo-holomorphic curves (14). We shall devote our future publication to these subjects.…”

“…Now we connect this picture with the previous field theory considerations. We recall that the center of the solitonic string is defined by M i = 0, so at this point the first Seiberg-Witten equation gives F +µν = ω µν thus, by (14), also F +µν /|ω| = 1 2 t +µν . Inserting this back to (24) and identifying |ω| y = Y | M i b=0 = ω we realize that the latter coincides with field theory transformation (4) taken at the string world-sheet.…”

Lectures on Symplectic Geometry

“…CP 2 #CP 2 #CP 2 is almost complex (proved by a computation with characteristic classes), but is neither complex (since it does not fit Kodaira's classification of complex surfaces), nor symplectic (as shown by Taubes [97] in 1995 using Seiberg-Witten invariants).…”

Heegaard Diagrams and Holomorphic Disks

“…It should also be pointed out that a different approach to understanding gauge theory from a geometrical point of view has been adopted by Taubes [103], building on his fundamental earlier work relating the Seiberg-Witten and Gromov invariants of symplectic four-manifolds [100], [101], [102].…”