Introduction to Seiberg-Witten's Invariants, Part I: Theory of Spinors

Abstract

In 1994, Witten has defined a monopole invariant and he has shown the equivalence of this invariant with Donaldson's polynomial using his result in \( \SS \)-duality. This new invariant is very powerful because the gauge group is abelian. By using such an invariant, many new results are found in the smooth, Kähler and even the symplectic categories. However, almost every paper in this topic write the monopole equations in a different way. Therefore is it necessary to clarify the basic idea behind the definition of such an invariant. In this paper we investigate the algebraic structure (Clifford algebra and \(\spinc\) representation ) underlying this invariant and explain the equations explicitly, especially the Kählerian case. Details of the computations are shown explicitly, and some minute mistakes in the existing papers are corrected.