Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

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150the composition of u, viewed as a linear map, with itself (i.e., in a coordinatesystem, (wow),^ = J^W^ij).kìFrom (8), one easily sees that the formula for the scalar curvature of gis s = s — \\u\ 2 , hence that the formula for the Lagrangian S appears indeed asthe sum of the Lagrangian S plus the energy term coming from the connection.Hence, one can derive the precise version of Kaluza's discovery (alsothe content of Kobayashi's remark) from (8). Physicists like to see this asa superposition of energies for the extended system (often called minimalcoupling).THEOREM 3. - On the total space M of an S 1 -bundle p: M —• M, theRiemannian metric g = p*g + a® à associateci with the S 1 -connection form awith curvature u is Einstein for the Constant X if and only if:i) LJ is a harmonic 2-form with Constant length 2vÀ ;ii) r = Xg — -ut ow.CiREMARK. - Because of i), the Einstein Constant X is necessarily positiveunless the bundle is fiat in which case g is Rìcci-fiat. The equations we havebeen writing hold equally well in the Lorentzian case. We come up with thesame constraint on the sign of A provided the fìbres are space-like (as Kaluzaassumed). Hence, we see that the vacuum Einstein equations force the bundleto be triyial, and also, what makes matters worse, the electromagnetic field tovanish identically. This phenomenon is often referred to as the Kaluza-Kleininconsistency.Condition ii) does not imply that the base metric is Einstein. It isnevertheless under this special assumption that explicit éxamples were foundby Kobayashi in [10]. If r = Xg, then we must haveuow = 2(A — X)g ,so that, if A ^ 0, the curvature form must be non degenerate. This is ofcourse the case if (A/, g) is Kàhlerian with a Kàhler form being a multiple ofthe curvature form u of the bundle. A typical example of this situation isM = being then a generatorof i/ 2 (M,Z). In that case, the manifold M is a standard sphere 5 2m+1 . Theconstruction can be carried over with other Hermitian symmetric irreductiblespaces. The main draw-back of these éxamples is that the metrics obtainedturn out to be homogeneous isotropy irreducible, hence belong to a well knownfamily.
In the hands of McKenzie Wang and W. Ziller (cf. section IV and[14]), this construction provides an infinite family of new examples of Einsteinmetrics provided one does noi take the base metric to be itself Einstein.151III. Non-abeiian b un die theory.a) The Kaluza-Klein approach to gauge theoriesIn 1954, C.N. Yang and R.L. Mills proposed in [15] a variational approachas classical model for strong interactiòns (the ones responsible for thecohesion of the nucleus). Later, this model was also used to describe ci assi -cally weak interactiòns (connected to the /?-decay). This proposai was knownto be only a part of the story since, due to their ranges, these interactiònshave to be dealt with quantum mechanically.This variational theory turns out to be identical to the theory of connectionson G-bundles, as was (much) later recognized. The group G to beconsidered here is the group on invariance of the interactiòns, as of today, U\for electromagnetism, S£/ 2 for weak interactiòns, SU3 for strong interactiòns.An elementary particle subject to a given interaction being attached to anirreducible representation of the invariance group, these groups can be determinedexperimentally by letting interact different particles subject to a giveninteraction, and comparing the outcome of the interaction with the tables ofrepresentations of the presupposed symmetry group.A compact Lie group G being given, we consider p : M —• M a (7-bundle.The fìeld to be extremized in this setting is a G— connection a for the Yang-Mills functional yM defìned as(il) yM( a ) = \j |n°p^where Q a denotes the curvature of the connection a. The volume element v 9refers to a Riemannian metric g supposedly given on the space-time manifoldM. Due to the use of this model as a first perturbation term towards aQuantum theory, it is indeed a Riemannian metric that physicists pick.How does the Kaluza-Klein Ansatz interfer with this problematic? Thiswas first contemplated by B. De Witt in 1964, and later developed by A.Trautman and R. Kerner (cf. [8]). One more time, one can play the game ofusing the data to contruct a metric on the total space ofthe bundle. This canbe done by using the same recipe as before, namely by setting(12) 9=P*9 + 9
Page 1: Rend. Sem. Mat. Univ. Poi. TorinoFa
Page 4 and 5: 146having the charge density as tim
Page 6 and 7: 148This approach immediately raised
Page 10 and 11: 152where g is a chosen G-invariant
Page 12 and 13: 154REMARK. - Before going on workin
Page 14 and 15: 156the base. By using (19), orie se
Page 16 and 17: 158the Hopf fibration S 15 -+• Ì
Page 18 and 19: 160dimensionai fibre. Following the
Page 20 and 21: 162We hope to have by now convinced

Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

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