^{JGP - Vol. 1,
n. 2, 1984}

^{Symmetry
of Einstein-Yang-Mills systems and dimensional reduction}

^{A}^{.
}^{JADCZYK}

^{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.}

1. INTRODUCTION

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 [1], 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 [2], [3]. [*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. [4], also [5] 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 [6] and Klein [7]. 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 [8], where the
sphere S^{2 } 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

2.2. Examples

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

3.6. Comments

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

5.1. Example

5.2. Group action on a principal bundle

5.3. Inner symmetries

5.4. Killing vectors of a connection

5.5. Example

6. DIMENSIONAL REDUCTION OF EINSTEIN-YANG-MILLS SYSTEMS

6.1. Application of the Reduction Theorem

6.2. Geometry of scalar fields

6.3. Results

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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This article is based on lectures given by the Author during the Trimester on Mathematical Physics at the Stefan Banach International Mathematical Centre, Warsaw Sept.-Nov. 1983.

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