Multiple Instantons and Characteristic Classes
The Hodge theorem states that, on a compact oriented manifold, each de Rham cohomology class can be represented by a harmonic form. Such a result has an important parallel in the Yang—Mills theory: each second Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. The purpose of this chapter is to establish the general theorem that, for each m = 1, 2,..., a similarly defined 2m-th cohomology class on S 4m generalizing the Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. In §3.1 we review some basic facts in 4 dimensions. In §3.2 we solve the Liouville equation. In §3.3 we present the explicit solutions of Witten in 4 dimensions based in the solution of the Liouville equation. In §3.4 we discuss the problem in all 4m dimensions and state our general representation theorem. In §3.5–§3.7 we prove the theorem.
KeywordsWeak Solution Gauge Field Characteristic Classis Topological Charge Liouville Equation
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