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AdSCFT Duality and Conformality for Non-Abelian Manifold

AdSCFT Duality and Conformality for Non-Abelian Manifold
AdSCFT Duality and Conformality for Non-Abelian Manifold

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International Journal of Modern Physics A,f c World Scienti?c Publishing Company AdS/CFT DUALITY AND CONFORMALITY FOR NON-ABELIAN ORBIFOLD PAUL H.FRAMPTON Institute of Field Physics,Department of Physics and Astronomy,Chapel Hill,NC 27599-3255An outline of the conformality approach to the gauge hierarchy is given including the use of non-abelian orbifolds to give uni?ed models of the left-right type.One can identify four approaches to the hierarchy problem by di?erent new (untested)physics at a TeV scale:?Supersymmetry.GUT uni?cation works better with TeV supersymmetry.?Technicolor.New strong dynamics at a TeV scale.?Large extra dimensions at a TeV scale.?Conformality at a TeV scale.Here,I discuss the fourth approach.It is motivated physically by the notion that at a TeV scale the standard model appears almost conformally invariant in the sense that the masses of the particles,as well as the QCD and weak scales appear almost vanishingly small.But the theory is not conformal invariant as it stands because the couplings still run.The conformality idea is to enrich the spectrum just so that the couplings cease to run at an infra-red ?xed point of the renormalization group.Nevertheless as will become clear there can still be a gauge coupling uni?cation at the TeV scale in a larger gauge group.The disparity of the 321couplings then arises from the group theory of the embedding in the larger group.

Using AdS/CFT duality,one arrives at a class of gauge ?eld theories of spe-cial recent interest.The simplest compacti?cation of a ten-dimensional super-string on a product of an AdS space with a ?ve-dimensional spherical manifold leads to an N =4SU (N )supersymmetric gauge theory,well-known to be con-formally invariant 1.By replacing the manifold S 5by an orbifold S 5/Γone arrives at less supersymmetries corresponding to N =2,1or 0depending on whether Γ?SU (2),SU (3),or ?SU (3)respectively,where Γis in all cases a subgroup of SU (4)~SO (6)the isometry of the S 5manifold.

It was conjectured in 2that such SU (N )gauge theories are conformal in the N →∞limit.In 3it was conjectured that at least a subset of the resultant

1

2AdS/CFT Duality and Conformality for Non-Abelian Orbifold

nonsupersymmetric N=0theories are conformal even for?nite N.Some?rst steps to check this idea were made in4.Model-building based on abelianΓwas studied further in5,6,7,arriving in7at an SU(3)7model based onΓ=Z7which has three families of chiral fermions,a correct value for sin2θand a conformal scale ~10TeV.

The case of non-abelian orbifolds bases on non-abelianΓhas now been studied8. We have considered all non-abelian discrete groups of order g<32.These are described in detail in9,10.There are exactly45such non-abelian groups.Because the gauge group arrived at by this construction5is?i SU(Nd i)where d i are the dimensions of the irreducible representations ofΓ,one can expect to arrive at models such as the Pati-Salam SU(4)×SU(2)×SU(2)type11by choosing N=2and combining two singlets and a doublet in the4of SU(4).Indeed we?nd that such an accommodation of the standard model is possible by using a non-abelianΓ.

The procedures for building a model within such a conformality approach are: (1)ChooseΓ;(2)Choose a proper embeddingΓ?SU(4)by assigning the compo-nents of the4of SU(4)to irreps ofΓ,while at the same time ensuring that the6 of SU(4)is real;(3)Choose N,in the gauge group?i SU(Nd i).

We choose N=2and aim at the gauge group SU(4)×SU(2)×SU(2).To obtain chiral fermions,it is necessary5that the4of SU(4)be complex4=4?.Actually this condition is not quite su?cient to ensure chirality in the present case because of the pseudoreality of SU(2).We must ensure that the4is not pseudoreal.

This last condition means that many of our45candidates forΓdo not lead to chiral fermions.For example,Γ=Q2n?SU(2)has irreps of appropriate dimensionalities for our purpose but it will not sustain chiral fermions under SU(4)×SU(2)×SU(2)because these irreps are all,like SU(2),pseudoreal.?Applying the rule that4must be neither real nor pseudoreal leaves a total of only19possible non-abelian discrete groups of order g≤31.The smallest group which avoids pseudoreality has order g=16but gives only two families.The technical details of our systematic search iare in8.

Here we mention only the simplest interesting non-abelian case which has g=24 and gives three chiral families in a Pati-Salam-type model11.

The?rst group that can lead to exactly three families occurs at order g=24 and isΓ=Z3×Q where Q(≡Q4)is the group of unit quarternions which is the smallest dicyclic group Q2n.

There are several potential models due to the di?erent choices for the4of SU(4) but only the case4=(1α,1′,2α)leads to three families.

Since Q×Z3is a direct product group,we can write the irreps as r i?αa where r i is a Q irrep andαa is a Z3irrep.We write Q irreps as1,1′,1′′,1′′′,2while the irreps of Z3are all singlets which we call1,α,|α?1.Thus Q×Z3has twelve irreps in all and the gauge group will be of Pati-Salam type for N=2.

If we wish to break all supersymmetry,the4may not contain a singlet ofΓ.Due to permutational symmetry among the singlets it is su?ciently general to choose4

AdS/CFT Duality and Conformality for Non-Abelian Orbifold3 =(1αa1,1′αa2,2αa3)with a1=0.

To?x the a i we note that the scalar sector of the theory which is generated by the6of SU(4)can be used as a constraint since the6is required to be real.This leads to a1+a2=?2a3(mod3).Up to permutaions in the chiral fermion sector the most general choice is a1=a3=+1and a2=0.Hence our choice of embedding is

4=(1α,1′,2α)(1) with

6=(1′α,2α,2α?1,1′α?1)(2) which is real as required.

We are now in a position to summarize the particle content of the theory.The fermions are given by

4×R I(3)

I

where the R I are all the irreps ofΓ=Q×Z3.

The scalars are given by

6×R I(4)

I

As described in more detail in8the scalars are su?cient to break the starting gauge symmetry SU(4)3×SU(2)12to the required4-2-2left-right structure,and with precisely three chiral families in16-plets.

1.S.Mandelstam,Nucl.Phys.B213,149(1983).

2.J.Maldacena,Adv.Theor.Math.Phys.2,231(1998).

3.P.H.Frampton,Phys.Rev.D60,041901(1999).

4.P.H.Frampton and W.F.Shively,Phys.Lett.B454,49(1999).

5.P.H.Frampton and C.Vafa,hep-th/990322

6.

6.P.H.Frampton,Phys.Rev.D60,085004(1999).

7.P.H.Frampton,Phys.Rev.D60,121901(1999).

8.P.H.Frampton and T.W.Kephart,Phys.Lett.B485,403(2000)and UNC-Chapel

Hill Report IFP-780-UNC(in preparation).

https://www.sodocs.net/doc/a0616471.html,eful sources of information on the?nite groups include:

D.E.Littlewood,it The Theory of Group Characters and Matrix Representations of

Groups(Oxford1940);

M.Hamermesh,Group Theory and Its Applications to Physical Problems(Addison-Wesley,1962);

J.S.Lomont,Applications of Fimite Groups(Academic,1959),reprinted by Dover (1993);

A.D.Thomas and G.V.Wood,Group Tables(Shiva,1980).

10.P.H.Frampton and T.W.Kephart,Int.J.Mod.Phys.A10,4689(1995).

11.J.C.Pati and A.Salam,Phys.Rev.D10,275(1974).

R.N.Mohapatra and J.C.Pati,Phys.Rev.D11,566(1975).

R.N.Mohapatra and G.Senjanovic,Phys.Rev.D12,1502(1975)

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