Principal bundle

The bundle Vk is the vector bundle associated with the principal bundle... 

SU(2) principal bundles. There is an additional Lagrangian for the 4 scalar field [93] ... 

Here G"v and F v are elements of the field strength tensors for the two SU(2) principal bundles. So far the theory is entirely parallel to the basic standard model of electroweak intereactions. In further work the Dirac and Yukawa Lagrangians that couple the Higgs field to the leptons and quarks will be included. It will then be pointed out how this will modify the B field. The < )4 field may be written according to a small displacement in the vacuum energy ... 

This non-Abelian gauge theory satisfies the usual transformation properties. If Jl is the base manifold in four dimensions, then the gauge theory is determined by an internal set of symmetries described by a principal bundle. Let Ua, where a = 1,2,be an atlas of charts on the Ji. The transitions from one chart to another is given by gap f/p —> Ua, where these determine the transition functions between sections on the principal bundle. The transform between one section to another is given by... 

This chapter is in no way intended to be a comprehensive introduction to the subject of principal bundles and characteristic classes. Rather, we have tried to assemble a number of relevant results to entice the reader to consult the standard texts. Most of the given proofs are sketchy, though the proofs of the statements that are needed for later applications are complete. 

Principal Bundles and Spaces with a Free Group Action... 

We are now ready to state what is often referred to as the main structure theorem for principal bundles. 

Clearly, the same combinatorial data describes all possible Z2-principal bundles. Therefore, we can freely switch between these two families of bundles. Starting from the line bundle, the associated Z2-principal bundle is obtained by replacing each line by the unit sphere S°. Conversely, starting from a Z2-principal bundle, the associated line bundle is obtained by sticking a line into each fiber, with this line oriented away from the identity element toward the nontrivial element of Z2. 

We refer the reader to the wonderful book of tom Dieck, [tD87], for further details on equivariant maps and associated bundles. We also recommend the classical book of Milnor and Stasheff, [MSta74], as an excellent source for the theory of characteristic classes of vector bundles. Generalities on bundles, including principal bundles, can be found in [Ste51]. 

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