Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 7574.14.10 SW Monopole Equations and Donaldson TheoryDevelopments in the understanding of N = 2 supersymmetric YM theory in4D suggest a new point of view about Donaldson theory [Donaldson (1990);Donaldson (1986); Donaldson (1987); Donaldson and Kronheimer (1990)] offour manifolds: instead of defining 4–manifold invariants by counting SU(2)instantons, one can define equivalent 4–manifold invariants by countingsolutions of a nonlinear equation with an Abelian gauge group. This is a‘dual’ equation in which the gauge group is the dual of the maximal torusof SU(2). This new viewpoint, proposed by Witten in [Witten (1994)],suggests many new results about the Donaldson invariants.Let X be an oriented, closed 4–manifold on which we pick a Riemannianstructure with metric tensor g. Λ p T ∗ X, or simply Λ p , will denote thebundle of real–valued p−forms, and Λ 2,± will be the sub–bundle of Λ 2consisting of self–dual or anti–self–dual forms.The monopole equations relevant to SU(2) or SO(3) Donaldson theorycan be described as follows. If w 2 (X) = 0, then X is a spin manifold andone can pick positive and negative spin bundles S + and S − , of rank two. 31In that case, introduce a complex line bundle L; the data in the monopoleequation will be a connection A on L and a section M of S + ⊗ L. Thecurvature 2–form of A will be called F or F(A); its self–dual and anti–self–dual projections will be called F + and F − .If X is not spin, the S ± do not exist, but their projectivizations P S ±do exist (as bundles with fibers isomorphic to CP 1 ). A Spin c structure(which exists on any oriented four-manifold can be described as a choiceof a rank two complex vector bundle, which we write as S + ⊗ L, whoseprojectivization is isomorphic to P S + . In this situation, L does not exist asa line bundle, but L 2 does; 32 the motivation for writing the Spin c bundle asS + ⊗L is that the tensor powers of this bundle obey isomorphisms suggestedby the notation. For instance, (S + ⊗ L) ⊗2 ∼ = L 2 ⊗ (Λ 0 ⊕ Λ 2,+ ). The data ofthe monopole equation are now a section M of S + ⊗ L and a connection onS + ⊗ L that projects to the Riemannian connection on P S + . The symbolF(A) will now denote 1/2 the trace of the curvature form of S + ⊗ L.Since L 2 is an ordinary line bundle, one has an integral cohomologyclass x = −c 1 (L 2 ) ∈ H 2 (X, Z). Note that x reduces modulo two to w 2 (X);31 If there is more than one spin structure, the choice of a spin structure will not matteras we ultimately sum over twistings by line bundles.32 One might be tempted to call this bundle L and write the Spin c bundle as S + ⊗L 1/2 ;that amounts to assigning magnetic charge 1/2 to the monopole and seems unnaturalphysically.