Introduction to Taubes’ Theorem

Abstract

At this stage our moduli space \( \hat{M} \), although by now a smooth orientable manifold, may still be empty, if there are no reducible connections! A theorem of Clifford Taubes [T] rules out this gloomy possibility. He establishes the existence of self-dual connections on a 4-manifold M whose intersection form is positive definite. Taubes’ Theorem complements work of Atiyah, Hitchin, and Singer [AHS], who construct moduli spaces for a more restricted class of manifolds. For these “half-conformally flat” manifolds, twistor theory can be used to convert Yang-Mills into a problem in algebraic geometry. In particular, the self-dual Yang Mills equations are well understood on S4 (with the standard metric), although the topology of the moduli space for k > 2 is not completely known. Our 4-manifold M is not in general half-conformally flat, and other methods are required. Taubes uses analytic techniques to build self-dual connections on M from the solutions on S4. The k = 1 instantons on S4 have a center b ∈ S4 and a scale ⋋ ∈ R+. As ⋋ → 0 the instanton becomes localized near b. One can imagine a limiting connection at → = 0 whose curvature is supported at b. Taubes grafts the localized self-dual connections onto M, where they pick up a small anti-self-dual curvature, and for → sufficiently small he perturbs them slightly to obtain self-dual connections. There results a family of instantons on M, parametrized by (0, ⋋0) x M, and in a later chapter we prove that these essentially form a collar of M in M, the limiting connections ⋋ = 0 being adjoined to form the compactification M = M. U M. KeywordsModulus SpaceConformal InvariancePrincipal BundleUnique ContinuationTwistor TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.