Communications in Mathematical Physics

, Volume 278, Issue 2, pp 307–327 | Cite as

Yang-Mills Detour Complexes and Conformal Geometry



Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.


Vector Bundle Einstein Metrics Einstein Manifold Twistor Operator Formal Adjoint 
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • A. Rod Gover
    • 1
  • Petr Somberg
    • 2
  • Vladimír Souček
    • 2
  1. 1.Department of MathematicsThe University of AucklandAuckland 1New Zealand
  2. 2.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic

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