Yang-Mills Detour Complexes and Conformal Geometry |
| |
Authors: | A Rod Gover Petr Somberg Vladimír Souček |
| |
Institution: | (1) Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand;(2) Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic |
| |
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 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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|