Group actions on principal bundles and dimensional reduction

✍ Scribed by J. Harnad; S. Shnider; J. Tafel

• Publisher: Springer
• Year: 1980
• Tongue: English
• Weight: 278 KB
• Volume: 4
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00417502

✦ Synopsis

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation. Invariance conditions for gauge fields under Lie transformation groups have been analyzed recently through the use of global geometrical [1,2] and local Lie algebraic [3] methods. In particular, the classification of group actions in principal bundles and the theorem of Wang [4] characterizing invariant connections have been generalized to include non-transitive group actions provided the orbit structure is sufficiently regular [2]. Similar results concerning infinitesimal invariance conditions expressed in local terms have been obtained by Forgacs and Manton [3], who further analyzed the implications of reducing the gauge field Lagrangian by such symmetries.