JGP - Vol. 1, n. 2, 1984
Symmetry of Einstein-Yang-Mills systems and dimensional reduction
Institute of Theoretical Physics, University of Wroclaw 50-205 Wroclaw, Cybulskiego 36, Poland
Abstract. The following topics are discussed: G-invariant Riemannian metrics and principal connections, dimensional reduction of Einstein and Yang-Mills systems, curvature of coset spaces, dimensional reduction of spinors, geometrical interpretation of color and Higgs charges.
It is interesting to assume that space-time points are endowed with some internal structure. In modern language one assumes that our 4-dimensional space-time M is a base of a fiber bundle (E,p,M). E is a multidimensional Universe (dim E = 4 + N), and p : E ->M is a projection map identifying points in E which we do not discriminate. The idea that the events we normally perceive are only shadows, or projections, of things which take place in much more dimensions can be attributed to Plato. The fact that (under normal conditions) we are perfectly blind to the extra dimensions is naturally expressed by assuming that the fibers of E are homogeneous spaces. In this way certain symmetry group G is introduced, and it is tempting to connect this group with internal symmetry groups which are so helpful for classifying of elementary particles multiplets. An example of this type of a structure is given by a gauge theory when formulated in terms of fiber bundles. One starts there with a principal bundle p : P -> M, and the fibers of P are group manifolds. In electromagnetism (G = U(1)) the extra dimension is related to an unobservable phase of a wave function. For non-Abelian gauge fields one still talks of a non-observable, non-integrable, phase factor , but it is no longer connected with representing of quantum mechanical states by rays (1-dimensional subspaces) rather than vectors. The attempts of introducing quaternionic Hilbert spaces, which are natural for G = SU(2), produced so far no workable model, maybe because of the rigidity of our thinking. In this connection see , . [Note: as of June 2, 2000, there has been some progress in this direction - see references in a recent paper by G. Emch and this author]. Although principal bundles proved to be an indispensable concept in discussing gauge fields and their interaction with matter, many people felt uneasy about an ad hoc introduction of such a very special geometrical structure. One way of a more general, and more natural, introducing of a fibration is by a dynamical mechanism called a spontaneous compactification (see e.g. , also  and references there). Exact geometrical meaning of this mechanism is not yet clear, and I shall focus here on a simpler idea which relates the fibration of E to a global action of some internal symmetry group G. A dynamical origin of G and its action on E is left open here. Keeping in mind the obvious shortcomings of our model it is nevertheless worthwhile to study it as a straightforward generalization of the principal bundle structure which proved to be already useful. Before we go into the details let us give first some relevant references. A unification of gravitation and electromagnetism (U(1) gauge field) based on the idea of a five-dimensional Universe was worked out by Kaluza  and Klein . A possibility of a non-Abelian generalization of this idea was discussed several times [8, 9, 10] and its full geometrical and dynamical content has been given in [11, 12, 13, 14]. In all these papers it was always assumed that E is a principal bundle, i.e. that the internal spaces are group manifolds. The only exception is the Souriau paper , where the sphere S2 was proposed as a model for an internal space related to the isospin group G = SU(2). A general framework of G-invariant dimensional reduction described below has been given by Coquereaux and Jadczyk [15 ].
For the convenience of the reader we include a selection of references (Ref. [35 - 48]) where a broader spectrum of problems and approaches to gauge fields and Kaluza-Klein theories is discussed.
2. MULTIDIMENSIONAL UNIVERSE AND ITS BUNDLE STRUCTURE
2.1. Assumptions and notation
2.3. Bundle structure of E
2.4. K = N|H as the authomorphism group of H\G
2.5. Local product representation of E
3. RIEMANNIAN GEOMETRY OF G/H
3.1. Lie algebra decomposition
3.2. The canonical moving frame
3.3. Invariant spinors
3.4. Curvature and Killing vectors
3.5. Curvature of H\G
4. G-INVARIANT DIMENSIONAL REDUCTION OF METRIC
4.1. Reduction theorem
4.2. The adapted moving frame
4.3. The Levi-Civita connection
4.4. Ricci and scalar curvature
5. SYMMETRIES OF GAUGE FIELDS
5.2. Group action on a principal bundle
5.3. Inner symmetries
5.4. Killing vectors of a connection
6. DIMENSIONAL REDUCTION OF EINSTEIN-YANG-MILLS SYSTEMS
6.1. Application of the Reduction Theorem
6.2. Geometry of scalar fields
7. COMMENTS ON RELATED TOPICS
7.1. Einstein-Cartan theory and spinors
7.2. S-invariant spinors
7.3. Color and Higgs charges
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