Symmetries of Einstein-Yang-Mills fields and dimensional reduction
The author studies the geometrical structure given by a metric and a Yang-Mills field on a manifold E invariant under the action of a compact Lie group S and with a gauge group R. Notice that E can be written locally as \(M\times S/I\), M being the manifold or orbits (space-time) and I a typical isotropy group for the S-action. Thus they show that there is a one to one correspondence between such structures and quadruples \((\gamma_{\alpha \beta},A^ a_{\beta},\phi^ m_{\beta},h_{\beta \alpha})\) of fields defined solely on M; \(\gamma_{\alpha \beta}\) is a metric on M, \(h_{\alpha \beta}\) are scalar fields characterizing the geometry of the orbits (integral spaces), \(\phi^ m_{\alpha}\) are other scalar fields (Higgs fields) characterizing the S-invariance of the Lie(R)-valued Yang-Mills field and \(A^ a_{\beta}\) is a Yang-Mills field for the gauge group N(I)/I\(\times Z(\mu (I))\), N(I) being the normalizer of I in S, \(\mu\) is a homomorphism of I into R associated to the S-action, and Z(\(\mu\) (I)) is the centralizer of \(\mu\) (I) in R. Then the Einstein-Yang-Mills Lagrangian of E is expressed in terms of the component fields on M. Examples and model building recipes are given.
scientific article; zbMATH DE number 3947143