Ivancevic_Applied-Diff-Geom

726 Applied Differential Geometry: A Modern Introductionequation just says that the spinor section ψ has to be in the kernel of theDirac operator. The second equation describes a relation between the self–dual part of the curvature associated to the connection A and the sectionψ in terms of the Clifford action.The mathematical setting for Witten’s gauge theory is considerably simplerthan Donaldson’s analogue: first of all it deals with U(1)−principalbundles (Hermitian line bundles) rather than with SU(2)−bundles, andthe Abelian structure group allows simpler calculations; moreover the equation,which plays a role somehow analogous to the previous anti–self–dualequation for SU(2)−instantons (see [Donaldson and Kronheimer (1990)]),involves Dirac operators and Spin c −structures, which are well known andlong developed mathematical tools (see [Roe (1988)] or [Libermann andMarle (1987)]).The main differences between the two theories arise when it comes tothe properties of the moduli space of solutions of the monopole equation upto gauge transformations. The SW invariant, which depends on the Chernclass of the line bundle L, is given by the number of points, counted withorientation, in a zero–dimensional moduli space.In Witten’s paper [Witten (1994)] the monopole equation is introduced,and the main properties of the moduli space of solutions are deduced.The dimension of the moduli space is computed by an index theory technique,following an analogous proof for Donaldson’s theory, as in [Atiyahet. al. (1978)]; and the circumstances under which the SW invariants providea topological invariant of the four–manifold are illustrated in a similarway to the analogous result regarding the Donaldson polynomials.The tool that is of primary importance in proving the results about themoduli space of Abelian instantons is the Weitzenböck formula for the Diracoperator on the Spin c −bundle S + ⊗ L: such a formula is a well known (see[Roe (1988)]) decomposition of the square of the Dirac operator on a spinbundle twisted with a line bundle L.A first property which follows from the Weitzenböck formula is a boundon the number of solutions: the moduli space is empty for all but finitelymany choices of the line bundle L.Moreover, as shown in [Kronheimer and Mrowka (1994a)], the modulispace is always compact: a fact that avoids the complicated analytic techniquesthat were needed for the compactification of the moduli space ofSU(2)−instantons (see [Donaldson and Kronheimer (1990)]).Another advantage of this theory is that the singularities of the modulispace (again this is shown in [Kronheimer and Mrowka (1994a)]) only ap-