Chapter

Instantons and Four-Manifolds

Volume 1 of the series Mathematical Sciences Research Institute Publications pp 28-43

The Yang-Mills Equations

  • Daniel S. FreedAffiliated withDepartment of Mathematics, University of Texas at Austin
  • , Karen K. UhlenbeckAffiliated withDepartment of Mathematics, University of Texas at Austin
  • , Mathematical Sciences Research InstituteAffiliated with

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Abstract

From now on we deal with a certain orbit space M. This “moduli space” is the set of solutions to the self-dual Yang-Mills equations, but divided out by a natural equivalence. The self-dual Yang-Mills equations are an elliptic (except along orbits of the group of gauge transformations) system of partial differential equations for a connection on a vector bundle over a smooth 4-manifold. The group of gauge transformations is the group of natural equivalences in the vector bundle, hence the natural equivalence in the problem, and so it acts on the space of solutions to the self-dual Yang-Mills equations.