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ISSN: 2090-0902
Journal of Physical Mathematics
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A Hierarchy of Symmetry Breaking in the Nonsymmetric Kaluza-Klein (Jordan-Thiry) Theory

Kalinowski MW*

Bioinformatics Laboratory, Medical Research Centre, Polish Academy of Sciences, Poland

*Corresponding Author:
Kalinowski MW
Bioinformatics Laboratory, Medical Research Centre
Polish Academy of Sciences, Poland
Tel: 61641643156
E-mail: [email protected]

Received Date: September 05, 2015; Accepted Date: January 13, 2016; Published Date: January 20, 2016

Citation: Kalinowski MW (2016) A Hierarchy of Symmetry Breaking in the Nonsymmetric Kaluza-Klein (Jordan-Thiry) Theory. J Phys Math 7:152. doi:10.4172/2090-0902.1000152

Copyright: © 2016 Kalinowski MW. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The paper is devoted to the hierarchy of a symmetr y breaking in the Non symmetric Kaluza–Klein (Jordan–Thiry) Theory. The basic idea consists in a deformation of a vacuum states manifold to the cartesian product of vacuum states manifolds of every stage of a symmetry breaking .In the paper we consider a pattern of a spontaneous symmetry breaking including a hierarchy in the Non symmetr ic Kaluza–Klein (Jordan–Thiry) Theory.

Introduction

In this paper we consider hierarchy of symmetry breaking in the Nonsymmetric Kaluza–Klein Theory and the Nonsymmetric Kaluza– Klein Theory with a spontaneous symmetry breaking and Higgs’ mechanism . In the second section we consider a Nonsymmetric Kaluza–Klein Theory and the Nonsymmetric Kaluza–Klein Theory with a spontaneous symmetry breaking and Higgs’ mechanism [1-6]. In the third section we develop a hierarchy of the symmetry breaking in our theory. For further development of the nonsymmetric Kaluza– Klein (Jordan–Thiry) Theory [7-10].

Elements of the Nonsymmetric Kaluza–Klein Theory in general non-Abelian case and with spontaneous symmetr y breaking and Higgs’ mechanism

Let equationbe a principal fiber bundle over a space-time E with a structural group G which is a semisimple Lie group. On a space-time E we define a nonsymmetric tensor equationsuch that

g=det(gμv)≠0

equation (2.1)

g[μv] is called as usual a skewon field (e.g., in NGT, [6,11-13] We define on E a nonsymmetric connection compatible with gμv such that

equation (2.2)

where equation is an exterior covariant derivative for a connection equation and equation is its torsion. We suppose also

equation (2.3)

We introduce on E a second connection

equation (2.4)

such that

equation (2.5)

equation (2.6)

Now we turn to nonsymmetric metrization of a bundle equation. We define a nonsymmetric tensor γ on a bundle manifold P such that

equation (2.7)

where π is a projection from P to E . On equation we define a connection ω (a 1-form with values in a Lie algebra g of G. In this way we can introduce on P (a bundle manifold) a frame equation such that equation

Thus our nonsymmetric tensor looks like

equation,A,B=1,2…..,n+4, (2.8)

equation (2.9)

where equation is a bi invarian t Killing–Cartan tensor on G and equation is a rightinvarian t skew- symmetric tensor on G, equation

We have

equation (2.10)

Thus we can write equation (2.11)

equation (2.12)

(equationare structural constant s of the Lie algebra g).

equation is the symmetr ic part of γ and equation is the anti symmetr ic part of γ We have as usual

equation (2.13)

and

equation (2.14)

is a curvature of the connection ω

equation (2.15)

The frame θA on P is partially nonholonomic. We have

equation (2.16)

Even if the bundle equation is trivial, i.e. for Ω=0 This is different than in an electromagnetic case explanied by Kalinowski MW [3]. Our nonsymmetric metrization of a principal fiber bundle gives us a rightinvarian t structure on P with respect to an action of a group G on P [3]. Having P nonsymmetric ally metrized one defines two connection s on P right- invarian t with respect to an action of a group G on P. We have

equation (2.17)

In our left horizontal frame θA

equation (2.18)

equation (2.19)

where D is an exterior covariant derivative with respect to a connection equation on P and equation its torsion. One can solve Equation (2.18)– (2.19) getting the following results

equation (2.20)S

where equation is an inverse tensor of g(α)

equation (2.21)

equation is an Ad-type tensor on P such that

equation (2.22)

equation is a connection on an internal space (typical fiber) compatible with a metric equation such that

equation (2.23)

equation (2.24)

and of course equation where is a torsion of the connection equation We also introduce an inverse tensor of g(α)

equation (2.25)

We introduce a second connection on P defined as

equation (2.26)

equation is a horizontal one form

equation (2.27)

equation (2.28)

In this way we define on P all analogues of four- dimension al quantities from NGT [6,11]. It means, (n+4) dimension al analogues from Moffat theory of gravitation, i.e. two connection s and a nonsymmetric metric equation .Those quantities are right- invarian t with respect to an action of a group G on P. One can calculate a scalar curvature of a connection equation getting the following result [1-3].

equation (2.29)

Where

equation (2.30)

is a Moffat–Ricci curvature scalar for the connection equation equation is a is a Moffat–Ricci curvature scalar for the connection equation, and equation is a Moffat–Ricci curvature scalar for the connection equation

equation (2.31)

equation (2.32)

Usually in ordinary (symmetric) Kaluza–Klein Theory one hasequation where GN is a Newtonian gravitational constant and c is the speed of light. In our system of units this is the same as in Non symmetric Kaluza–Klein Theory in an electromagnetic case [3,4] In the non- Abelian Kaluza –Klein Theory which unifies GR and Yang–Mills field theory we have a Yang–Mills lagrangian and a cosmological term. Here we have

equation (2.33)

and equation plays a role of a cosmolog ical term.In order to incorporate a spontaneous symmetr y breaking and Higgs’ mechanism in our geometrical unification of gravitation and Yang–Mills’ fields we consider a fiber bundle equation over a base manifold equation where E is a space-time, equation equation are semisimple Lie groups. Thus we are going to combine a Kaluza–Klein theory with a dimension al reduction procedure.

Let equation be a principal fiber bundle over V=E × M with a structural group H and with a projection π, where

equation is a homogeneous space, G is a semisimple Lie group and Go its semisimple Lie subgroup. Let us suppose that (V, γ) is a manifold with a nonsymmetric metric tensor

equation (2.34)

The signature of the tensor γ is equation Let us introduce a natural Phenomenon

equation (2.35)

It is convenient to introduce the following notation. Capital Latin indices with tilde equation run 1,2,3…,m+4 ,equation. Lower Greek indices α,β,γ,δ=1,2,3,4 and lower Latin indicesequation Capital Latin indicesequation. Lower Latin indices with tilde equation The symbol over θA and other quantities indicates that these quantities are defined on V. We have of courseequation where equation equation, equation.

On the group H we define a bi- invarian t ( symmetr ic) Killing– Cartan tensor

equation (2.36)

We suppose H is semisimple, it means equation.We define a skew- symmetr ic right- invarian t tensor on H

equation

Let us turn to the nonsymmetric metrization of equation .

equation (2.37)

where

equation (2.38)

is a nonsymmetric right-invarian t tensor on H.One gets in a matrix form (in the natural frame (2.35))

equation (2.39)

equation equation and real, then

equation (2.40)

The signature of the tensor k is equation. As usual, we have commutation relations for Lie algebra of H, h

equation (2.41)

This metrization of equation is right- invarian t with respect to an action of H on P.Now we should nonsymmetric ally metrize M=G/G0. M is a homogeneous space for G (with left action of group G). Let us suppose that the Lie algebra of G, g has the following reductive decomposition

equation (2.42)

where g0 is a Lie algebra of G0 (a subalgebra of g) and m (the complement to the subalgebra g0) is AdG0 invarian t, + means a direct sum. Such a decomposition might be not unique, but we assume that one has been chosen. Sometimes one assumes a stronger condition for m, the so called symmetr y requirement

equation (2.43)

Let us introduce the following notation for generators of g:

equation (2.44)

This is a decomposition of a basis of g according to (2.42). We define a symmetr ic metric on M using a Killing–Cartan form on G in a classical way. We call this tensor h0. Let us define a tensor field equation on G/G0, equation using tensor field h on G. Moreover, if we suppose that h is a bi invarian t metric on G (a Killing–Cartan tensor) we have a simpler construction.The complement m is a tangent space to the point {εG0} of M, ε is a unit element of. We restrict h to the space m only. Thus we have h0{εG0} at one point of M. Now we propagateequation sing a left action of the group G equation equation }) is of course AdG0 invarian t tensor defined on m andequation

We define on M a skew- symmetr ic 2-form k0. Now we introduce a natural frame on M. Let equation be structure constant s of the Lie algebra g, i.e

equation (2.45)

Yj are generators of the Lie algebra g. Let us take a local section s: equation of a natural bundle equation where.equation The local section s can be considered as an introduction of a coordinate system on M.

Let ωMC be a left- invarian t Maurer–Cartan form and let

equation (2.46)

Using de composition (2.42) we have

equation (2.47)

It is easy to see that equation is the natural (left- invarian t) frame on M and we have

equation (2.48)

equation (2.49)

According to our notation equation

Thus we have a nonsymmetric metric on M

equation (2.50)

Thus we are able to write down the nonsymmetric metric on equation

equation (2.51)

where

equation

equation

equation

equation

equation The frame equation is unholonomic:

equation (2.52)

where equation are coefficient s of nonholonomicity and depend on the point of the manifold M=G/G0 (they are not constant in general). They depend on the section s and on the constantsequation We have here three groups H,G, G0 H,G, G0. Let us suppose that there exists a homomorphism μ between G0 and H, μ(G0)

equation (2.53)

such that a centralizer of μ(G0) in H, Cμ is isomorphic to G. Cμ, a centralizer of μ(G0) in H, is a set of all element s of H which commute with element s of μ(G0)), which is a subgroup of H. This means that H has the following structure, equation

equation (2.54)

If μ is a iso morphi sm between G0 andμ(G0) one gets

equation (2.55)

Let us denote by μ ′ a tangent map to μ at a unit element. Thus μ ′ is a differential of μ acting on the Lie algebra element s. Let us suppose that the connection ω on the fiber bundle P is invarian t under group action of G on the manifold equation. According to Kobayashi [14-17] this means the following.

Let e be a local section of equationequation and equation Then for every g∈G there exists a gauge transformation ρg such that

equation (2.56)

f *means a pull-back of the action f of the group G on the manifold V. According to Hlavaty [13-25] we are able to write a general form for such an ω. Following [17] we have

equation (2.57)

(An action of a group G on V=E × G/G0 means left multiplication on a homogeneous space M=G/G0.) where equation are components of the pull-back of the Maurer–Cartan form from the de composition (2.47) equation is a connection defined on a fiber bundle Q over a space-time E with structural group Cμ and a projection πE . Moreover, Cμ=G and equation is a 1-form with values in the Lie algebra g.

This connection describes an ordinary Yang–Mills’ field gauge group Cμ=G on the space-time E. Φ is a function on E with values in the spaceequation of linear maps

equation (2.58)

satisfying Φ

equation (2.59)

Thus

equation (2.60)

Let us write condition (2.57) in the base of left-invarian t form equation which span respectively dual spaces to g0 and m [24,25]. It is easy to see that

equation (2.61)

and

equation (2.62)

From (2.59) one gets

equation (2.63)

where equation are structure constant s of the Lie algebra g and equation are structure constant s of the Lie algebra h Equation (2.63) is a constraint on the scalar field equation.For a curvature of ω one gets

equation (2.64)

Thus we have

equation (2.65)

equation (2.66)

equation (2.67)

where equation means gauge derivative with respect to the connection equation defined on a bundle q over a space-time E with a structur al group G

equation (2.68)

equation is the curvature of the connection equation in the base {Yi}, generators of the Lie algebra of the Lie group G g, equation is the matrix which connects {Yi} with { Xc } . Now we would like to remind that indices a,b,c refer to the Lie algebra h, equation to the space m (tangent space to M), equation to the Lie algebra g0 and i,j,k to the Lie algebra of the group G, g. The matrix equation establishes a direct relation between generators of the Lie algebra of the subgroup of the group H iso morphi c to the group G.

Let us come back to a construction of the Nonsymmetric Kaluza– Klein Theory on a manifold P. We should define connection s. First of all, we should define a connection compatible with a nonsymmetric tensor γΑΒ, Equation (2.51)

equation (2.69)

equation (2.70)

equation

where equation is the exterior covariant derivative with respect to equation and equation its torsion. Using (2.51) one easily finds that the connection (2.69) has the following shape

equation (2.71)

where equation γ is a connection on the space-time E and on the manifoldequationwith the following properties .

equation (2.72)

equation

equation (2.73)

equation (2.74)

(equation is an exterior covariant derivative with respect to a connection equation is a tensor of torsion of a connection equation is an exterior covariant derivative of a connection equation and equation its torsion.On a space-time E we also define the second affine connection such that

equation (2.75)

equation

We proceed a nonsymmetric metrization of a principal fiber bundle equationaccording to (2.51). Thus we define a right-invarian t connection with respect to an action of the group H compatible with a tensor equation

equation (2.76)

where equation D is an exterior covariant derivative with respect to the connection equation and equation its torsion. After some calculations one finds

equation (2.77)

Where

equation (2.78)

equation (2.79)

Ldc is Ad-type tensor with respect to H (Ad- covariant on equation

equation (2.80)

equation (2.81)

equation (2.82)

We define on P a second connection

equation (2.83)

Thus we have on P all (m+4) dimension al analogues of geometrical quantities from NGT, i.e. equation

Let us calculate a Moffat–Ricci curvature scalar for the connection equation

Hierarchy of a Symmetry Breaking

Let us incorporate in our scheme a hierarchy of a symmetry breaking. In order to do this let us consider a case of the manifold

equation (3.1)

where

equation (3.2)

equation (3.3)

equation (3.4)

Every manifold Mi is a manifold of vacuum states if the symmetr y is breaking from Gi+1 to Gi, Gk=G.

Thus

equation (3.5)

We will consider the situation when

equation ( 3.6)

This is a constraint in the theory. From the chain (3.5) one gets

equation (3.7)

and

equation (3.8)

The relation (3.6) means that there is a diffeomorphism g onto G/ G0 such that

equation (3.9)

This diffeomorphism is a deformation of a product (3.1) in G/G0. The theory has been constructed for the case considered before with G0 and G. The multiplet of Higgs’ fields Φ breaks the symmetr y from G to G0 (equivalently from G to G0 in the false vacuum case). gi mean Lie algebras for groups Gi and mi a complement in a decomposition (3.8) . On every manifold Mi we introduce a radius ri (a “size” of a manifold) in such a way that equation. On the manifold G/G0 we define the radius r as before. The diffeomorphism g induces a contragradient transformation for a Higgs field Φ in such a way that

equation (3.10)

The fields Φi, i=0,…,k-1.

this way we get the following decomposition for a kinetic part of the field Φ and for a potential of this field:

equation (3.11)

equation (3.12)

where

equation (3.13)

equation (3.14)

equation (3.15)

Equation (3.5) is a non symmetr ic tensor on a manifold Mi.

equation (3.16)

equation are structure constant s of the Lie algebra gi . The scheme of the symmetr y breaking acts as follows from the group Gi+1 to Gi (Gt) (if the symmetr y has been broken up to Gi+1). The potential equation has a minimum (global or local) for equation k = 0,1. The value of the remaining part of the sum (3.12) for fields Φj, j < i , is small for the scale of energy is much lower ( rj >ri , j < i ). Thus the minimum of equation is an approximate minimum of the remaining part of the sum (3.12)). In this way we have a descending chain of truncations of the Higgs potential. This gives in principle a pattern of a symmetr y breaking. However, this is only an approximate symmetr y breaking. The real symmetr y breaking is from G to G0 (or to G0 in a false vacuum case). The important point here is the diffeomorphism g.

equation (3.17)

equation (3.18)

The shape of g is a true indicator of a reality of the symmetr y breaking pattern. If

g = Id +δ g (3.19)

where δg is in some sense small and Id is an identity, the sums (3.11)- (3.12) are close to the analogous formulae from the expanation of Kalinowski [5,10]. The smallness of g is a criterion of a practical application of the symmetr y breaking pattern (3.5) . It seems that there are a lot of possibilities for the condition (3.9). Moreover, a smallness of δg plus some natural conditions for groups Gi can narrow looking for grand unified models. Let us notice that the decomposition of M results in decomposition of cosmolog ical terms

equation (3.20)

where

equation (3.21)

where equation is a non symmetr ic connection on Mi compatible with the non symmetr ic tensor equation and equation its curvature scalar. The truncation procedure can be proceeded in several ways. Finally let us notice that the energy scale of broken gauge bosons is fixed by a radius ri at any stage of the symmetr y breaking in our scheme.

Let us consider Equation (3.10) in more details. One gets

equation (3.22)

where

equation (3.23)

equation (3.24)

is a matrix of Higgs’ fields transformation.

According to our assumptions one gets also:

equation (3.25)

For g is an invertible map we have det g* ( y) ≠ 0 .

We have also

equation (3.26)

and

equation (3.27)

equation (3.28)

equation (3.29)

such that

equation (3.30)

equation (3.31)

For an inverse tensor equation one easily gets

equation (3.32)

We have

equation (3.33)

In this way we have for the measure

equation (3.34)

where

equation (3.35)

equation (3.36)

In the case of equation one gets

equation (3.37)

where

equation (3.38)

where

equation (3.39)

Moreover, to be in line in the full theory we should consider a chain of groups Hi i = 0,1, 2,..., k −1 , in such a way that

equation (3.40)

For every group Hi we have the following assumptions

equation (3.41)

and Gi+1 is a centralizer of Gi in Hi. Thus we should have

equation (3.42)

We know from elementary particles physics theory that

equation

equation

and that G2 is a group which plays the role of H in the case of a symmetr y breaking from equation to Uel (1) . However, in this case because of a factor U(1), M=S2. Thus M0 =S2 and G2⊂H0.

It seems that in a reality we have to do with two more stages of a symmetr y breaking. Thus k=3. We have

equation (3.43)

equation (3.44)

equation

equation (3.45)

and

equation (3.46)

equation (3.47)

equation (3.48)

and

equation (3.49)

equation (3.50)

We can take for G, SU(5), SU(10), E6 or SU(6). Thus there are a lot of choices for G2, H1 and H.We can suppose for a trial that

equation (3.51)

(3.50)

equation (3.52)

Thus

equation (3.53)

We can try with F4= H0

In the case of H

equation

Thus we can try with E7, E8

equation

equation (3.55)

In this way we have

equation

Thus we can try with

equation (3.57)

But in this case

equation

This seems to be nonrealistic. For instance, if G= SO(10), E6,

equation

In this case we get

equation

And H could be SO(10), SO(18), SO(20).In this approach we try to consider additional dimensions connecting to the manifold M more seriously, i.e. as physical dimensions, additional space-like dimensions. We remind to the reader that gauge-dimensions connecting to the group H have different meaning. They are dimensions connected to local gauge symmetr ies (or global) and they cannot be directly observed. Simply saying we cannot travel along them. In the case of a manifold M this possibility still exists. However, the manifold M is diffeomorphically equivalent to the product of some manifolds Mi, i = 0,1, 2,..., k −1, with some characteristic sizes ri The radii ri represent energy scales of symmetr y breaking. The lowest energy scale is a scale of weak interactions (Weinberg–Glashow–Salam model) equation cm. In this case this is a radius of a sphere S2 The possibility of this “travel” will be considered in the concept explanied by Kalinowski [26]. In this case a metric on a manifold M can be dependent on a point x∈E (parametrically).It is interesting to ask on a stability of a symmetr y breaking pattern with respect to quantum fluctuations. This difficult problem strongly depends on the details of the model. Especially on the Higgs sector of the practical model. In order to preserve this stability on every stage of the symmetr y breaking we should consider remaining Higgs’ fields (after symmetry breaking ) with zero mass. According to S. Weinberg, they can stabilize the symmetry breaking in the range of energy

equation (3.58)

i.e. for a symmetry breaking from Gi+1 to Gi

It seems that in order to create a realistic grand unified model based on non symmetr ic Kaluza–Klein (Jordan–Thiry) theory it is necessary to nivel cosmolog ical terms. This could be achieved in some models due to choosing constant s ξ and ζ and μ . After this we can control the value of those terms, which are considered as a selfinteraction potential of a scalar field Y. The scalar field Y can play in this context a role of a quintessence .

Let us notice that using the equation

equation (3.59)

and (3.27) one gets

equation (3.60)

In this way we get constraints for Higgs’ fields, Φ0, Φ1, Φk-1

equation

Solving these constraints we obtain some of Higgs’ fields as functions of in dependent components [26]. This could result in some cross terms in the potential (3.12) between Φ’s with different i. For example a term

equation

where Φ′ means in dependent fields. This can cause some problems in a stability of symmetry breaking pattern against radiative corrections. This can be easily seen from Equation (3.59) solved by in dependent Φ′ ,

Φ = BΦ′ (3.61)

equation (3.62)

Where B is a linear operator transforming in dependent Φ′ into Φ.

We can suppose for a trial a condition similar to (3.59) for every i = 0,..., k −1 ,

equation (3.63)

where equation are structure constant s for the Lie algebra hi of the group Hi equation are structure constant s of the Lie algebra gi+1 equation are indices belonging to Lie algebra gi and equation to the complement mi.

In this way

equation (3.64)

In this case we should have a consistency between (3.63) and (3.60) which impose constraints on C,f μ and equation where refer to equation Hi, Gi+1. Solving (3.63) via introducing in dependent fields Φi′ one gets

equation (3.65)

Combining (3.62) , (3.64) , (3.65) one gets

equation (3.66)

Equation (3.66) gives a relation between in dependent Higgs’ fields Φ′and Φ ′i. Simultaneously it is a consistency condition between Equation (3.59) and Equation (3.63). However, the condition (3.63) seems to be too strong and probably it is necessary to solve a weaker condition (3.60) which goes to the mentioned terms V( Φi′, Φ′j) . The conditions (3.63) plus a consistency (3.66) avoid those terms in the Higgs potential. This problem demands more investigation. ϕ (g)= {gG1}

It seems that the condition (3.9) could be too strong. In order to find a more general condition we consider a simple example of (3.5). Let G0= {e} and K=2 In this case we have

equation (3.67)

equation (3.68)

equation (3.69)

In this way G1× G/G1 is diffeomorphically equivalent to G. Moreover, we can consider a fibre bundle with base space G /G1 and a structural group G1 with a bundle manifold G. This construction is known in the theory of induced group representation done by Trautman [27]. The projection ϕ :G® G /G1is defined by equation. The natural extension of (3.69) is to consider a fibre bundle (G, G/ G1, G1). In this way we have in a place of (3.69) a local condition

equation (3.70)

where U⊂ G/G1 is an open set. Thus in a place of (3.9) we consider a local diffeomorphism

equation (3.71)

where

equation

Ui Mi , i = 0,1, 2,...., k −1 , are open sets. Moreover we should define projectors i ϕ , i = 0,1, 2,...., k −1 ,

where

equation (3.72)

i.e.

equation (3.73

equation

in a unique way. This could give us a fibration of G /G0 in equation

For gGi+1 we simply define

equation (3.74)

If equation we define

equation (3.75)

Thus in general

equation (3.76)

where

equation (3.77)

Thus in a place of (3.9) we have to do with a structure

equation (3.78)

such that

equation (3.79)

where

equation (3.80)

This generalizes (3.9) to the local conditions (3.71). Now we can repeat all the considerations concerning a decomposition of Higgs’ fields using local diffeomorphisms gU ( gU*) in the place of g (g*). Let us also notice that in the chain of groups it would be interesting to consider as G2

equation (3.81)

suggested by Salam and Pati, where SU(4) unifies equation This will be helpful in our future consideration concerning extension to super symmetr ic groups, i.e. U(2,2) which unifies equation to the super Lie group U(2,2) considered by Mohapatra. Such models on the phenomenological level incorporate fermions with a possible extension to the super symmetric SO(10) model. They give a natural framework for lepton flavour mixing going to the neutrino oscillations incorporating see-saw mechanism for mass generations of neutrinos. In such approaches the see-saw mechanism is coming from the grand unified models. Our approach after incorporating manifolds with anticommuting parameters, super Lie groups, super Lie algebras and in general supermanifolds (superfibrebundles) can be able to obtain this. However, it is necessary to develop a formalism (in the language of supermanifolds, superfibrebundles, super Lie groups, super Lie algebras) for non symmetr ic connections, non symmetr ic Kaluza– Klein (Jordan–Thiry) theory. In particular we should construct an analogue of Einstein–Kaufmann connection for supermanifold, a non symmetr ic Kaluza–Klein (Jordan–Thiry) theory for superfibrebundle with super Lie group. In this way we should define first of all a non symmetr ic tensor on a super Lie group and afterwards a non symmetr ic metrization of a superfibrebundle. Let us notice that on every stage of symmetr y breaking, i.e. from Gi+1to Gi, we have to do with group Gt (similar to the group G0). Thus we can have to do with a true and a false vacuum cases which may complicate a pattern of a symmetr y breaking.

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