Notes on Seiberg–Witten Gauge Theory

In the fall of 1994 E. Witten announced a “new gauge theory of 4-manifolds”, capable of giving results analogous to the earlier theory of Donaldson (see [13]), but where the computations involved are “at least a thousand times easier” (Taubes). The new theory begins with the introduction of the monopole equations, whose physical motivation lies in the new results in N = 2 supersymmetric Yang-Mills theory announced in [41], [42]. The equations are in terms of a section of a Spinor bundle and a U(1) connection on a line bundle L. The first equation just says that the spinor section ψ has to be in the kernel of the Dirac operator. The second equation describes a relation between the self-dual part of the curvature associated to the connection A and the section ψ in terms of the Clifford action. The mathematical setting for Witten’s gauge theory is considerably simpler than Donaldson’s analogue: first of all it deals with U(1)-principal bundles (hermitian line bundles) rather than with SU(2)-bundles, and the abelian structure group allows simpler calculations; moreover the equation, which plays a role somehow analogous to the previous anti-self-dual equation for SU(2)-instantons (see [13]), involves Dirac operators and Spinc-structures, which are well known and long developed mathematical tools (see [40] or [28]). The main differences between the two theories arise when it comes to the properties of the moduli space of solutions of the monopole equation up to gauge transformations. The Seiberg–Witten invariant, which depends on the Chern class of the line bundle L, is given by the number of points, counted with orientation, in a zero-dimensional moduli space. The present introduction to the subject of Seiberg–Witten gauge theory will be taken mainly from the paper by Witten [52]; however extensive use will be made of other references that have recently appeared.