Introduction to Taubes’ Theorem

  • Daniel S. Freed
  • Karen K. Uhlenbeck
  • Mathematical Sciences Research Institute
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 1)

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 bS4 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.

Keywords

Manifold Expense Kato Karen 

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Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Daniel S. Freed
    • 1
  • Karen K. Uhlenbeck
    • 1
  • Mathematical Sciences Research Institute
    • 2
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.BerkeleyUSA

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