2006/06/30 by A. Rod Gover, Petr Somberg, Vladimı́r Souček +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Conformal field theory #Conformal geometry #Conformal map #Connection (principal bundle) #Differential operator #Gauge theory #Geometric Analysis and Curvature Flows #Geometry #Holonomy #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Pure mathematics #Signature (topology) #Twistor theory #Vector bundle #Yang–Mills existence and mass gap #gr-qc #hep-th #math-ph #math.DG #math.MP #msc:53A30 #msc:53A55
paper · pdf · doi:10.1007/s00220-007-0401-5
published as Commun.Math.Phys.278:307-327,2008 · Final version. To appear: Communications of Mathematical Physics. 23 pages
openalex publication_date 2008/01/07 · arxiv created 2008/03/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 bundle connection respects a metric on the vector 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.