Class. Quantum Grav. 1(1984) 517-530. Printed in Great Britain
Colour and Higgs charges in G/H Kaluza-Klein theory
A Z Jadczyk*
International Centre for Theoretical Physics, Trieste, Italy
* On leave of absence from Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, 50-205 Wroclaw, Poland.
Received 24 April 1984
Abstract. Geodesics in a multidimensional Universe with G-invariant metric are studied and differential equations describing their space-time projection are derived. It is shown that, when internal spaces are cosets rather than group manifolds, then, in addition to the colour charge through which the particle interacts with the gauge field, a new charge arises which couples to the scalar fields only. This new charge, called the Higgs charge, is shown to be of a nonlinear nature. For certain cosets it may contribute to the colour charge non-conservation.
1. Introduction
Recently a geometrical theory of dimensional reduction was developed based on the concept of G-invariance of a Riemannian metric in a multidimensional Universe [1]. In the present paper geodesics in a multidimensional Universe are studied, and it is shown that the components of the velocity pointing into the extra dimensions give rise to the coloured and higgsonic charges. From the formal point of view our equations (5.1)-(5.3) generalise the well known Kerner-Wong equation (see [2-5], also [6] where motion of a charged string is treated) in two ways: firstly, the internal space is taken to be a homogeneous space G/H rather than a group manifold, and secondly, interaction with the Jordan-Thierry scalars, originating from the metric on G/H, is taken into account. Before going into the details of this paper's results, it is worthwhile analysing the relation between the geometrical framework used in [1], and the, so-called, Kaluza-Klein theory. The Kaluza-Klein mechanism is supposed to work as follows (see e.g. [7, 8], and references therein): one starts with a generally covariant field theory in (4 + n) dimensions (e.g. eleven-dimensional supergravity), containing among its fields {phi} the metric field gAB (A, B = 1, 2,..., 4 + n). Suppose the theory admits a ground state {phi0} with the property that the (4+n)-dimensional space E splits into a (local) product E = M x S with respect to the ground state metric g0AB(x, y) = (g0mu nu(x), galpha beta(y)), with x in M, y in S. If M is four-dimensional of signature ( + - - -), and if S is compact, then one says that a spontaneous compactification takes place. The next step to be taken is called dimensional reduction. It is believed that the lowest excitations from the ground state can be interpreted in terms of massless fields on M alone - the gravity, Yang-Mills fields, scalars. A general geometrical framework taking care of the whole mechanism has not yet been worked out, although much work has been recently done in two directions: (a) constructing particular models of physical interest (e.g. [8-11]), and (b) elaborating mathematical apparatus describing selected features of the models (e.g. [12-16]). Of particular interest are recent works on eleven-dimensional super-gravity, where it was found that a ground state need not have as its symmetry group G the maximal symmetry group allowed by the topology of the internal space S. For example, the ground state metric of the eleven-dimensional gravity may have only SO(5) x SU(2) (instead of SO(8)) symmetry ('squashed seven-sphere', [17]) or, another possibilty, the sphere can be metrically round, but the other fields may have non-zero values phi0<>0 in the ground state breaking the symmetry (from SO(8) down to SO(7) in [18], [19]). In such cases it is of interest to study those excitations of the ground state metric which have G as its symmetry group. It is this problem that was solved in [1] (see also [16]). Under assumption that the internal space is a homogeneous space G/H it was shown that the most general G-invariant metric on M x (G/H) is described in terms of metric gmu nu on M, gauge field Aalpha for the gauge group K = N(H)|H, and charged scalar fields galpha beta describing internal geometry of S = G/H. If H = {e} (i.e. if S is a group manifold) then K = G, but for, say, G = SO(8), H = SO(7) one finds K = Z2, so that K looks much smaller than one would expect. In general, therefore, the condition of G-invariance will select only certain and not all zero modes. Thus, the analysis of interaction of point particles with the massless fields gmu nu, Amu, and galpha beta , given in the present paper, should be extended when a geometry of a complete ansatz (see [7], [8]) is fully understood, so that the interaction with the remaining massless modes can be taken into account.
With the above in
mind, let us make some more comments on the results of this paper. It
is important to observe that the Lie-algebra decomposition
corresponding to a homogeneous space G/H is not Lie(G) = Lie(H)+S,
where S can be interpreted as the space tangent to G/H at
the origin, but a more subtle one: Lie(G) = Lie(H)+Lie(K)+L,
where Lie(K) is the Lie algebra of K = N(H)|H (the
effective gauge group), N(H) being the normaliser of H
in G. Consequently the metric galpha beta on S
= G/H splits into two kinds of Jordan-Thierry-type fields: gab (a,
b--the indices of L), and gab (ab -
the indices of K). It is shown that the interaction of the
particle with both fields is of the form Zalpha beta Dmu,galpha
beta, Zalpha beta being composite: Zalpha
beta = lalpha lbeta. la
is just the coloured charge, interacting with Amu,
while la is
a new charge, called a higgsonic charge. We give evolution equations
for all charges and argue that the Higgs charges are nonlinear: they
take values in the manifold L/Ad(H) of H-orbits in L (H
acting on L by the adjoint representation). Readers who are
not interested in the details may get the idea of the present paper
by reading §5 where the summary of results and a simple example
are given.
2.
Geometry of a G-Universe
3. Geodesics in G-Universe
4. Equations of
motion for a particle carrying coloured and higgsonic charges
5. Summary and example
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