Class. Quantum Grav. 3 (1986) 29-42
The Institute of Physics

Harmonic expansion and dimensional reduction in G/H Kaluza-Klein theories

Centre de Physique Thfiorique, Section 2, CNRS, Lumiuy, Marseille Cedex 2, France + CERN, Geneva, Switzerland

Abstract. We propose a geometrical framework for harmonic expansion and dimensional reduction of matter fields in Kaluza-Klein theories with the most general G-invariant ansatz. Generalised Peter-Weyl and Frobenius theorems provide a basis for harmonic expansion, and a mechanism is shown by which the dimensional reduction of matter fields is then automatically accomplished. In particular, we discuss the dimensional reduction of tensor and spinor fields, and of the Laplace and Dirac operators.

Harmonic expansion and dimensional reduction are two mechanisms used in discussing the effective four-dimensional content of higher-dimensional field theories admitting a spontaneous compactification. A general discussion of these problems was given by Salam and Strathdee [1] while Witten [2] gave a broad discussion of the harmonic expansion and dimensional reduction of spinor fields and, in particular, of the difficulties in obtaining the effective four-dimensional chiral asymmetry. Spacetime manifold was treated in those papers locally and no intrinsic geometrical meaning was given to the constructions. On the other hand, certain global aspects of the problem were studied by Romer [3] and Bleecker [4,4a]; however, both authors restricted their discussion to the case of internal space being a group manifold, and spinors were not discussed al all in [4, 4a]. Manton [5] considered dimensional reduction of fermions interacting with a Yang-Mills field. His discussion was, however, restricted to invariant spinors (corresponding to a being the trivial representation of G according to the notation of § 2) and product metric.

In this paper we consider a G/H Kaluza-Klein theory as formulated in [6]. As a geometrical model for the extended spacetime we take a manifold E on which a global symmetry group G acts (G can be thought of as the internal symmetry group of a ground state, but the scheme is broad enough to apply to any Riemannian manifold with a compact group of isometries; it will also apply to the still more general case of G being a local isometry group, as discussed in [6a]). Then E locally looks like a product M xS of spacetime M and internal homogeneous space S~G/H. The spacetime manifold is defined globally and unambiguously as the manifold of G orbits (internal spaces). What is not unique is a local product representation of E as M x G/H--any two such representations differ by a gauge transformation with gauge group N(H)IH (see [6, 7]). All the analysis given in this paper is directed by the aim not to distinguish any particular product representation of E (a global one may even not exist), and in doing so to exhibit an intrinsic geometrical meaning of the constructions.

The outline of the paper is as follows: we start with matter fields on E considered as sections of some equivariant vector bundle over E, i.e. of a vector bundle F= F( E; F) with base E and typical fibre F, with G acting on F by bundle automorphisms, The set GAM(F) of all cross sections of ff is the space of matter-field configurations in the extended spacetime E. We decompose GAM(F) according to the irreducible representations a of G and, for each a, define harmonics of type a. This constitutes the first step: harmonic expansion of fields (§ 2). Then, in § 3 we show that the space of harmonics of a given type a (which are still fields on E) can be interpreted as the space of sections of an appropriate 'effective' vector bundle Fa over M. This is the second step: dimensional reduction. After that we consider the particularly interesting situation when F has a structure group, say, R; or, in other words, when F is a bundle associated to some principal bundle U = U(E; R) via a certain representation rho of R on F. The global symmetry group G is assumed to act on U by automorphisms. We use the results of [8, 9] to find out the effective ('dimensionally reduced') gauge group P, and the effective principal bundle O(M; R) over M. We also find the effective representation pa, of R. on an appropriate Fp and show that the Fa can also be constructed as the bundle associated to the effective principal bundle U~ via pa,. In § 4 we apply these results to discuss the case of U being the bundle of orthonormal frames of E endowed with a G-invariant metric, and in § 5 we discuss tensor and spinor fields and dimensionally reduced Laplace and Dirac operators.

2. Harmonic expansion
3. Dimensional reduction of G-vector bundle
4. Dimensional reduction of the bundle of orthonormal frames
5. Dimensional reduction of tensor and spinor fields
5.1 Laplace operator on scalar fields
5.2 1-forms

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