Nuclear Physics B276 (1986) 617-628 North-Holland, Amsterdam









CERN. Geneva. Switzerland


Received 2 December 1985


We consider thc G-invariant Kaluza-Klein scheme on M G/H leading to the N(H)/H gauge group and demonstrate its consistency. The full-scale ansatz with G N(H)/H gauge bosens from M G/H compactification is argued to be, in general, inconsistent.


1. Introduction


The present paper is a continuation and, in a sense, also a closure of a series of papers [1-5] in which we investigated the geometrical meaning of "dimensional reduction" - a procedure for obtaining an effective four-dimensional multifield theory from a multidimensional uni- (or "few") field theory. In this series of papers, we have given geometrical foundations to a whole family of theories of the Kaluza-Klein type under the assumption that the internal spaces are orbits of a certain global isometry group G, thus being homogeneous spaces of the type G/H. In the simplest model of this type, one considers just one field in a multidimensional universe the metric tensor, with lagrangian (R - 2Lambda)sqrt(g). This simple Kaluza-Klein theory was investigated in detail in ref. [2], and it will be sufficient for our purposes to restrict our attention mainly to this model also in the present paper*. The message coming from the results of ref. [2] can be summarized as follows: consider spontaneous compactification on M S where the ground state has (internal) symmetry group G' acting transitively on S, and let G be a subgroup of G' such that S = G/H (i.e. G is transitive on S). Then, taking into account all G-invariant modes, and only those, of the metric in the (m + s)-dimensional universe E ~ M (G/H), M being m-dimensional space-time, one obtains all effective m-dimensional theory containing a metric tensor of M, a Yang-Mills field with gauge group N(H)/H, N(H) being the normalizer of H in G, and, also, a certain multiplet of scalar fields whose composition and colour content depend on G and on H. Now, in ref. [6] it has been shown, using general arguments, that a G-invariant ansatz should always be consistent provided that one takes into account all G-invariant modes. The discussion of the consistency problem given in refs. [6] and [7] (see also ref. [8] for a review of recent results) seems to indicate also that the most popular ansatz (see, for example, refs. [9 - 12]) involving Killing vectors, which is explicitly not G-invariant, should be considered as guilty of inconsistency unless proved innocent for a specific model**.

In the present paper, we tackle this consistency problem with the following results:

(i) We show by explicit calculations that dimensional reduction (from M G/H to M), based on the G-invariant ansatz for the metric tensor [2], and thus giving rise to Yang-Mills fields with gauge group N(H)/H, leads indeed to the effective m-dimensional lagrangian and field equations which are fully consistent with the original (m + s)-dimensional (s = dim(G/H)) theory.

(ii) On the other hand, following certain geometrical considerations of ref. [13], we consider another ansatz, also of a geometrical nature but more general than the first one, which gives rise to an effective m-dimensional theory with gauge group N(H)/H G (i.e. the maximal one allowed by the geometry of G-action). Here we argue that the resulting m-dimensional theory is, in general, inconsistent with the original one.

We wish to close this introductory section with a few comments:

(1) A Kaluza-Klein scheme is called "consistent" if (a) it admits a compactifying ground state solution: (b) if every solution of the resulting m-dimensional field equations can also be interpreted as a solution of the original (m + s)-dimensional theory***

(2) A scheme which is inconsistent need not necessarily be all wrong; however, such an inconsistency suggests that "something" goes wrong with the truncation scheme.

(3) A scheme which is consistent must still be shown to be energetically stable before being accepted as a candidate for a realistic theory. We do not mean that an inconsistent ansatz is physically meaningless and we do not intend to discuss here the possible physical relevance of such an ansatz.


* The several known cases of a consistent mm-G-invariant ansatz are discussed in ref. [8].

** Adding primary Yang-Mills fields to this scheme can be achieved using the results of refs. [3] and [4], and the addition also of antisymmetric tensor fields and spinors (as is necessary in supersymmetric models) can be carried out using ref. {5].

*** Notice that (a) is implied by (b); we are grateful to M. Duff for discussions of the consistency problem.


Note added

In ref. [2], the factor 1/2 in (3.5.7) should be 1/4 (as in the CERN preprint). Also, the term "isotropy" on p. 97 six lines from the bottom should be replaced by ad H.

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[1] R. Coquereaux and M.J. Duff, unpublished (1981)

12] R. Coquereaux and A. Jadczyk, Comm. Math. Phys. 90 [1983) 79

[3] A. Jadczyk and K. Pilch, Lett. Math. Phys. 8 (1984) 97

[4] R. Coquereaux and A. Jadczyk, Comm. Math. Phys. 98 (1985) 79

[5] R. Coquereaux and A. Jadczyk, Class. Quantum Gray. 2 (1985) 50

[6] M.J. Duff and C.N. Pope, Nucl. Phys. B255 (1985) 355

[7] M.J. Duff, B.E.W. Nilsson, C.N. Pope and N.P. Warner, Phys. Lett. 149B (1984) 90

[8] M.J. Duff, CERN preprint TH.4243/85 (1985)

[9] E. Witten, Nuch Phys. B186 11982) 412

[10] A. Salam and J. Strathdee, Ann, of Phys. 141 (1982) 316

[11] J. Strathdee, On Kaluza-Klein theories, in Unification of the fundamental particle interactions II, eds. J. Ellis and S. Ferrara (Plenum, 1983)

[12] C. Wetterich, Realistic Kaluza-Klein theories, in Perspectives in particles and fields, Cargese 1983, eds. M. Levy et al. (Plenum, New York, 1983)

[13] A. Jadczyk, preprint ITP UWr 85/642 (1985)

[14] S.W. Hawking and G.F.R. Ellis, The large-scale structure of space-time (Cambridge University Press, 1973)

[15] R. Coquereaux and A. Jadczyk, Geometry of hidden dimensions, eds. Bibliopolis, to be published
(note: this book on hyperdimensional physics has appeared witha different title: Riemannian geometry,...)



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