of the centralizer of ~ in ~. The one-dimensional subspace of the trivial representation. R~=C~ is spanned by the element ~=(dimR~)|R~|174 where i~| and iR~| ...
24. 25. 26. 27. 28.
A~ N. Sisakian, N. B. Skachkov, I. L. Solovtsov, and O. Yu. Shevchenko, J!NR Rapid Co~mun. No0 12-87, JINR, Dubna (1987), p. 14. B. M. Barbashov et al., Phys. Lett. B, 33, 484 (1970). A. N. Sisakyan, Tr. Fiz. Inst. Akad. Nauk No. 140, FIAN SSSR, Moscow (1970). N. B. Borisov and P. P. Kulish, Teor. Mat. Fiz., 51, 335 (1982). G~ S~ Iroshnikov, Yad. Fiz., 38, 512 (1983); 44, 1554 (1986).
SYMMETRIC SPACES AND HIGGS MODELS IN THE METHOD OF DIMENSIONAL REDUCTION. II.
THEORIES WITH ONE MULTIPLET OF SCALAR FIELDS I. Po Volobuev, Yu~ A. Kubyshin, and Zh. M. Mourao In the present paper, which is a continuation of [i], gauge theories obtained by dimensional reduction of multidimensional "free" gauge theories with symmetry are studied. An investigation of the properties of a definite class of subalgebras of the classical Lie algebras (including both regular and irregular subalgebras) yields a sufficient condition for the reduced theory to contain only one irreducible multiplet of scalar fields. Examples of the explicit construction of such theories for the case when the space of additional dimensions is the sphere S s are given.
i.
Introduction
In this paper we continue the study begun in [i] of gauge theories obtained by dimensional reduction of "free" gauge theories in multidimensional spaces of the form M = M 4 x g/H, where M ~ is Minkowski space, and G/H is a compact symmetric space. In the earlier paper we developed a general method for calculating the potentials of the scalar fields of the reduced theory in a form valid for all symmetric spaces G/H and all gauge groups K of the multidimensional theory. It was found that these potentials are potentials of Higgs type, leading to spontaneous synmletry breaking. However, in the general case the symmetry breaking corresponding to them can have a very complicated nature, since the theory can contain several irreducible multiplets of scalar fields. It is therefore of interest to find a class of multidimensional theories for which dimensional reduction leads to the simplest Higgs models with one irreducible multiplet of scalar fields; such models are of physical interest, and spontaneous symmetry breaking in them has been well studied. In addition, this class of multidimensional theories has a direct relation to the theory of spontaneous compactification [2]. We recall that the scalar fields of the reduced theory are described by a linear mapping @(x): ~ which intertwines the irreducible representation A d ( H ) ~ with the reducible representation Ad(T(H))~. Here, as in [i], ~, ~, and ~ denote the Lie algebras of the groups G, H, and K ~ = ~ @ ~ is the canonical decomposition of the symmetric Lie algebra [3]~ and the homomorphism on a special infinitedimensional lattice, this being a two-dimensional generalization of a Cayley tree. It is constructed by the successive addition of shells. As zeroth shell we take a single plaquette (simple two-dimensional square), and to each of its edges we add y new plaquettes, the complete set of these forming the first shell. The second shell is constructed by adding 7 plaquettes to each free edge of the first shell. Continuing this process, we obtain an infinitely branching surface whose Hausdorff dimension is dH = ~ (see Fig~ I, which shows the zeroth and first shells for 7 = 2). However, the model itself is studied only on a set of plaquettes that lie deep within the lattice. Such plaquettes are characterized solely by coordination number 7 + 1 and must be equivalent to each other, so that all the relations obtained below solely for the central plaquette are valid for them. We close all plaquettes at infinity, i.eo, on the given lattice the cohomology 2-group H2(K, G) is equal to infinity. We shall cavil such a lattice of plaquettes a Bethe lattice, and in this sense it differs from the 2 dimensional generalization of a Cayley tree. The terminology is taken from spin models~ in which one proceeds similarly (see [6], w and [7,8]). A further aspect should be noted. In the framework of the i/d expansion for the pure gauge Ising model a phase transition of the first kind has been obtained for d = ~ [9,10]. On the other hand, our lattice by itself does not have closed surfaces (cycles), and there, fore in models considered on it critical behavior must be absent in the case of free boundary conditions [3,11]. To avoid this trivial result, it is necessary to impose special boundary conditions, identifying all the boundary edges. However, in what follows w e s h a l l , for convenience, simply fix on the entire boundary the value u~ = +i of the variables, this being equivalent to halving the partition function. The influence of the boundary conditions on the critical behavior will be elucidated in Seco 3. Erevan Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 78, No. 2, pp. 281-288, February, 1989. Original article submitted July 9, 1987o
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0040-5779/89/7802-0200512.50 9 1989 Plenum Publishing Corporation
