Class. Quantum Grav. 3 (1986) 29-42

The Institute of Physics

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

R** **Coquereaux
and A Jadczyk

*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.

1. Introduction

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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References

[1] Salam A and Strathdee J 1982 *Ann. Phys., NY ***141**
316-52

[2] Witten E 1983 *Preprint *Princeton University

[3] Romer H 1982 *Preprint THEP 82/6 *Universitaet Freiburg

[4] Bleecker D 1983 *Int. J. Theor. Phys. ***22** 557-74

[4a] Bleecker D 1984 *Int. d. Theor. Phys. ***23** 735-50

[5] Manton N S 1981 *Nucl. Phys. ***B 193** 502-16

[6] Coquereaux R and Jadczyk A 1983 *Commun. Math. Phys. ***90**
79-100

[6a] Jadczyk A 1985 *Preprint ITP UWn *85/642

[7] Jadczyk A 1984 *J. Geom. Phys. ***I** 97-126

[8] Jadczyk A and Pilch K 1984 *Lett. Math. Phys. ***8**
97-104

[9] Coquereaux R and Jadczyk A 1985 *Cornmun. Math. Phys.
***98** 79-104

[10] Husemoller D 1975 *Fibre bundles *(Berlin: Springer)

[11] Segal G 1968 *Equivariant K-theory. IHES Publ. Math.
No 34 *129 51

[12] LashorR K 1982 *Ill. J. Math. *26257-71

[13] Wallach N R 1973 *Harmonic analysis on homogeneous spaces
*(New York: Marcel Dekker)

[14] Dieudonne J 1975 *Elements d'analyse *vol 5 (Paris:
Gautheir-Villars)

[15] Palla L 1984 *Z. Phys. ***C 24** 195 204

[16] Bott R 1965 in *Proc. Syrup. in Honor of Marston Morse
*ed S S Cairns (Princeton: Princeton University

Press)

[17] Kirillov A A 1976 *Elements of the Theory of Representations
*(Berlin: Springer)

[18] Briining J and Heintze E 1979 *Inv. Math. ***50**
169-203

[19] Forgacs P and Manton N S 1980 *Cornmun. Math. Phys. ***72**
15-35

[20] Hamad J, Schnider S and Vinet L 1980 *J. Math. Phys.
***21** 2719-24

[21] Hamad J, Schnider S and Tafel J 1980 *Lett. Math. Phys.
***4** 107-13

[22] Kobayashi S and Nomizu K 1963 *Foundations of Differential
geometry *vol I (New York: Interscience) [23] Dieudonne J 1971 *Elements
d'Analyse *vol 4 (Paris: Gauthier-Villars)

[24] Milnor J 1963 *L'Enseignement Math. *9 198-203

[25] Bichteler K 1968 *J. Math. Phys. *9 813-5

[26] Geroch R 1968 *J. Math. Phys. *9 1739-44

[27] Atiyah M and Hirzebruch F 1970 in *Essays on Topology
and Related Topics. Memoires dedies a Georges*

*de Rham *ed A Haefiiger and R Narasimhan (Berlin: Springer)

[28] Chichlinsky C 1972 *Trans. Am. Math. Soc. ***172**
307-15

[29] Hanori A 1978 *Inv. Math. ***48** 7 31

[30] Hawking S W and Pope C H 1978 *Phys. Lett. ***73B**
42-4

[31] Bott R 1963 in *Differential and Combinatorial Topology.
Proc. Symp. in Honor of Marston Morse *ed

S S Cairns (Princeton: Princeton University Press)

[32] Palais R 1965 in *Semina ron theAtiyah-Singer index theorem
*ed R Palais (Princeton: Princeton University

Press)

[33] Hitchin N 1974 *Adv. Math. ***14** 1-55

[34] Lawson H B Jr and Michelson M L 1980 *J. Diff. Geom.
*15 237-67

[35] Atiyah M F and Singer I M 1984 *Proc. Nat. Acad. Sci.
***81** 2597-600

[36] Parthasaraty K 1972 *Ann. Math. ***96** 1-30

[37] Kerner R 1981 *Ann. Inst. H Poincare 34 *437-63

.