Communications in
Mathematical Physics
90, 79-100 (1983)
© Springer-Verlag 1983
Abstract. Let G be a compact group of transformation (global symmetry group) of a manifold E (multidimensional universe) with all orbits of the same type (one stratum). We study G invariant metrics on E and show that there is one-to-one correspondence between those metrics and triples (g, A, h), where g is a (pseudo-) Riemannian metric on the space of orbits (space-time), A is a Yang-Mills field for the gauge group N|H, where N is the normalizer of the isotropy group H in G, and h are certain scalar fields characterizing geometry of the orbits (internal spaces). The scalar curvature of E is expressed in terms of the component fields on M. Examples and model building recipes are also given. The results generalize those of non-abelian Kaluza-Klein theories to the case where internal spaces are not necessarily group manifolds.
1. Introduction
First special, and later general, theories of relativity invoked a picture of the universe as being modelled on a four-dimensional space-time manifold. On the other hand, in order to describe regularities of discrete quantum numbers characterizing elementary particles, a concept of "internal" (as opposed to "external", i.e., space-time) symmetry, and with it that of internal space, was introduced. The idea behind what we call a "multidimensional universe" (denote it by E) is that external and internal spaces are nothing but two aspects of one geometrical entity E, and all elementary forces in nature should be but a reflection of a unique geometry. By some, not yet fully understood, mechanism, certain configurations of a simple, multidimensional field theory are distinguished, and give rise to a "spontaneous compactification" of extra dimensions (see [1] and references therein). This idea is at the root of the so-called "dimensional reduction." As a result, the multidimensional universe splits into a four-dimensional space-time M and a compact internal space S. At the same time, the original simple field(s) on E split(s) into components which are identified with the conventional fields on M, like scalars, tensors, Yang-Mills fields (in supergravity, also spinors), etc. In theories of the Kaluza-Klein type, the "simple theory" is a multidimensional gravitation, and the "distinguished configurations" are those Riemannian metrics on E which admit a given compact group G of isometrics. The case where the internal space S is a group manifold itself has been studied in many papers (see, e.g., [2-8]) and the geometrical structure of this kind of dimensional reduction is by now well understood. A G-invariant metric g_{A B} splits into a gravitational field g_{mu nu} on M, Yang-Mills field A-mu^i, with G as the gauge group, and scalar fields h_{i j}(x) describing the metric in the internal space (these are generalizations of Jordan-Thierry-Brans-Dicke scalars (see [4, 6, 7]). Recently, attempts have been made [9-13] to generalize these results to a case where S is not necessarily a group manifold, but rather a homogeneous space of the type G/H. No straightforward generalization of the original Kaluza-Klein idea has, however, been obtained (in particular, E of [12] is the associated bundle so that it hardly even makes sense to consider G invariance) and the construction given in the present paper may be thought of as an alternative to those of [12, 13]. Homogeneous spaces have also been used (see [14] and references therein) to provide solutions to supergravity theories (where there are matter fields besides the metric); however, the so-called "Kaluza-Klein ansatz" used in these papers is not, in general, G invariant.
In the present paper we solve the following general problem: what are the most general algebraic and geometric properties of an extended universe E under the only requirement that G (a compact group) be a group of internal (global) symmetries? In Sect. 2 we show that E can be written locally as the product M x S with S= G\H, and that a local symmetry group K=N|H arises in a natural way, N being the normalizer of H in G. In Sect. 3, we characterize all G invariant metrics on E and show that the local symmetry group K is at the same time the gauge group; we prove that there is one-to-one correspondence between G invariant metrics on E and triples (g, A, h), where g, is a metric on M, A are Yang-Mills fields corresponding to the gauge group K-N]H, and h are scalar fields. We also express the scalar curvature of E in terms of these fields [formulae (3.5.7) and (3.5.8)]. Examples (see Tables 1 and 2) are given in Sect. 4. We want to stress the fact that a principal bundle structure, so characteristic in mathematical descriptions of gauge fields, arises automatically and naturally in the present approach the principal bundle emerges as a specific submanifold of the extended universe E (see Fig. 2).
Besides its physical aspects and motivations this paper contains, uses, or refers to quite a number of mathematical techniques and results. Most of them are either standard or are simple exercises in differential geometry. However, we believe that the main results of Sect. 3 are new. The reader who is not interested in a "mathematical balast" may get the idea of the present paper by reading the summaries in Sects. 2.6 and 3.6, and also Sect. 4, where examples and model building recipes are discussed.
2. Bundle Structure of the Extended Space-Time
2.1. Bundle Structure of E
2.2. The Normalizar and the Local Symmetry Group
2.3. Construction of the Associated Bundle E(G\H,M)
2.4. Global Action of G on E(G\H,M)
2.5. Bundle Structure of E (cont.)
2.6. Summary
3. Metric and Curvature
3.1. Decomposition of Lie(G)
3.2. Adapted Basis
3.3. Vertical Moving Frame
3.4. Characterization of G Invariant Metrics on E
3.5. Curvature
3.6. Summary
4. Comments and Examples
4.1. Counting the Number of Scalar Fields
4.2. A Class of Almost Trivial Examples
4.3. Model Building
4.4. Comments
Download Paper in PDF format (930 kB)
References
1. Witten, E.: Search for a realistic Kaluza-Klein theory. Nucl. Phys. B186, 412 (1981)
2. Kerner, R.: Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group. Ann. Inst. H. Poincare 9, 143 (1968)
3. Cho, Y.M.: Higher-dimensional unifications of gravitation and gauge theories. J. Math. Phys. 16, 2029 (1975)
4. Cho, Y.M., Freund, P.G.O.: Non-abelian gauge fields as Nambu-Goldstone fields. Phys. Rev. D12, 1711 (1975)
5. Cho, Y.M., Jang, Pong Soo: Unified geometry of internal space with space-time. Phys. Rev. D12, 3789 (1975)
6. Scherk, J., Schwarz, John H.: How to get masses from extra dimensions. Nucl. Phys. B153, 61 (1979)
7. Orzalesi, C.A.: Multidimensional unified theories. Fortschr. Phys. 29, 413 (1981)
8. Coquereaux, R., Duff, M.J.: Unpublished
9. Souriau, J.M.: Relativite multidimensionnelle non stationaire. Colloque de Royaumont (1959) In: Les Theories relativistes de la gravitation. Paris: CNRS 1962
10. Domokos, G., Kovesi-Domokos, S.: Gauge fields on coset spaces. Nuovo Cimento 44A, 318 (1978)
11. Salam, A., Strathdee, J.: On Kaluza-Klein theory. Ann. Phys. 141, 316 (1982)
12. Percacci, R., Randjbar-Daemi, S.: Kaluza-Klein theories on bundles with homogeneous fibres. ICTP preprint ICTP/82/18 (1982)
13. Cho, Y,M.: Gauge theories on homogeneous fibre bundle. CERN preprint TH. 3414 (1982)
14. Duff, M.J.: Invited lecture at the third Marcel Grossman meeting, Shanghai (1982). CERN preprint TH. 3451 (1982)
15. Montgomery, D., Yang, C.T.: The existence of a slice. Ann. Math. 65, 108 (1956)
16. Montgomery, D., Yang, C.T.: Orbits of highest dimension. Trans. Am. Math. Soc. 87, 284 (1958)
17. Abraham, R., Marsden, J.E.: Foundations of mechanics (2nd edn.). London: Benjamin 1978
18. Borel, A., Bredon, G., Floyd, E.E., Montgomery, D., Palais, R.: Seminar on transformation groups, Ann. Math. Stud. Vol. 46. Princeton, New Jersey: Princeton University Press 1960
19. Hsiang, W.Y.: In: Proceedings of the Conference on Transformation Groups, New Orleans 1967, Mostert, P.S. (ed.). Berlin, Heidelberg, New York: Springer 1968
20. Dieudonne, J.: Treatise on analysis, Vol. III. New York, London: Academic Press 1972
21. Palais, S.R.: On the existence of slices for action of non compact Lie groups. Ann. Math. 73, 295 (1961)
22. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. I/II. New York, London: Interscience Publ. 1963
23. Lichnerowicz, A.: Geometry of groups of transformations. Leyden: Noordhoff Intern. Publ. 1977
24. Barut, A.O., Raczka, R.: Theory of group representations and applications. Warszawa: PWN-Polish Sci. Publ. 1980
25. Patera, J., Sankoff, D.: Tables of branching rules and representations of simple Lie algebras. Montreal: Les Presses de l'Universite de Montreal 1974
26. Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962
27. Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. 120, 59 (1968)
28. Poncet, J.: Groupes de Lie compacts de transformations. Commun. Math. Helv. 33, 109 (1959)
29. Berger, H.: Les varietes riemanniennes homogenes normales simplement connexes a courbure strictement positive. Ann. Scu. Norm. Sup. Pisa 15, 179 (1961)
30. Wolf, J.A.: Spaces of constant curvature. New York: McGraw-Hill 1967
31. Stora, R.: Yang Mills instantons, geometrical aspects. Erice Lectures (27 June - 9 July 1977)
32. Atiyah, M.F., Hitchin, N.Y., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. London A362, 425-461 (1978)
33. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 21 (1976)
34. Berard Bergery, L.: Sur la courbure des metriques riemanniennes invariantes des groupes de Lie et des espaces homogenes. Ann. Sci. Ecole Norm. Sup. 11, 543 (1978)
35. Jensen, G.R.: The scalar curvature of left-invariant Riemannian metrics. Ind. Univ. Math. J. 20. 1125 (1971)
36. Koiso, N.: Non-deformability of Einstein metrics. Osaka J. Math. 15, 419 (1979)
37. Nagano, T.: A problem on the existence of an Einstein metric. J. Math. Soc. Jpn. 19, 30 (1967)
38. Jensen, G.R.: Einstein mctrics on principal fiber bundles. J. Differential Geometry 8, 599 (1973)
39. Berger, M.: Rapport snr les varietes d'Einstein. In: Analyse sur les varietes. Metz 1979. Soc. Math. de France, Asterisque 80 (1980)
40. Bourguignon, J.P.: Deformations des metriques d'Einstein. In: Analyse sur les varietes. Metz 1979. Soc. Math. de France, Asterisque 80 (1980)
41. Awada, M.A., Duff, M.J., Pope, C.N.: N=8 supergravity breaks down to N= 1. Preprint 1CTP/82-82/4 (1982)
Communicated by G. Mack
Received January 10, 1983; in revised form February 16, 1983
* On leave of absence from Centre de Physique Theorique, F-13288 Luminy, Marseille, France ** On leave of absence from Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, PL-50-205 Wroclaw, Poland
*** Work done in the framework of the Project MRI. 7. of the Polish Ministry of Science, Higher Education and Technology
Links to my other online papers dealing with hyperdimensional physics
.