Yang-Mills detour complexes and conformal geometry

Abstract

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.

Final version. To appear: Communications of Mathematical Physics. 23 pages

References in corpus (6)

• Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus
• The Ambient Obstruction Tensor and Q-Curvature
• Almost conformally Einstein manifolds and obstructions
• The Normal Conformal Cartan Connection and the Bach Tensor
• Subcomplexes in Curved BGG-Sequences
• A sub-product construction of Poincare-Einstein metrics

Cited by in corpus (15)

• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Gravitational- and Self- Coupling of Partially Massless Spin 2
• Partial Masslessness and Conformal Gravity
• Conformal Field Theory and Algebraic Structure of Gauge Theory
• Detours and Paths: BRST Complexes and Worldline Formalism
• Renormalized Volume
• Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds
• BRST Detour Quantization
• Perturbative Quantum Field Theory and Homotopy Algebras
• Almost Einstein and Poincare-Einstein manifolds in Riemannian signature
• Stress and Geometry for Isotropic Singularities
• Quaternionic Kaehler Detour Complexes & N=2 Supersymmetric Black Holes
• On a new normalization for tractor covariant derivatives
• A conformally invariant Yang-Mills type energy and equation on 6-manifolds
• Bernstein-Gelfand-Gelfand resolutions for linear superalgebras