# orbifold compactification of heterotic string and 6D and flavor unification models

###### Abstract

A orbifold compactification of the heterotic string is
considered. The resulting 6D GUT groups can be or
plus some hidden sector groups.
The supersymmetry is reduced to
. In particular, the 6D model with one spinor
representation 128 can reduce to the previous
5D or family
unification models after compactifying the sixth dimension.
To obtain one spinor, we have
to take into account the left-over center of .
We also comment on the model.

[Key words: orbifold, superstring, 6D
SO(16) GUF, holonomy]

###### pacs:

12.10.-g, 11.25.Mj, 11.30.Hv, 11.30.Ly^{†}

^{†}preprint: SNUTP 02/020

## I Introduction

With the discovery of the top quark, the three family standard model has been filled and seems to be the theory below 100 GeV energy scale. Presumably, the observed weak CP violation is the most compelling reason for the three family standard model(SM), but it lacks the theoretical reasoning of why the same fermion representation repeats three times. In this regard, the grand unification of families or flavors(GUF) attempts to unify the SM gauge couplings without the repetition of fermionic representations of the grand unification gauge group[2], which seems to be the most promising rationale toward interpreting multi generations. In early days of GUF, indeed there appeared models without repeated fermionic representations[2, 3].

But with the advent of superstring models, the GUF idea seems to be automatically implemented in superstring models. In particular, the orbifold models usually give 4D supersymmetric(SUSY) three family models and seemed to be one of the fundamental principles of compactification of extra internal space[4, 5]. Some orbifold models are very close to the standard model[5], but in most cases, there appear charged SM singlet fermions, which may not be consistent with the observed weak mixing angle . Therefore, one of the key points to require in the compactification is not allowing unfamiliar fermions(UF). For example, vectorlike charged fermions are UF’s since they are not appearing in the SM. For this reason, the flipped [7] attracted a great deal of attention in the fermionic construction of 4D string models[8]. However, the orbifold compactification of seemed to be not easy. Recently, a 5D model without UF’s has been considered to give a 4D three family flipped model which does not violate major low energy observations[9, 10].

From a 10D superstring theory, one has to compactify six extra internal spaces to obtain a 4D SUSY theory[11]. Most widely considered superstring theory is the heterotic . But the field theoretic orbifold compactification[12] need the string compactification of the internal space smaller than six, so that we can consider SUSY field theory in . For the possibility of complexifying the internal space, we consider compactifying even dimensions, and hence our consideration of the internal space is four dimensions, leading to a 6D SUSY field theory model. The field theoretic compactification[12] need the string compactification of the internal space smaller than six, so that we can consider SUSY field theory in . For the possibility of complexifying the internal space, we consider compactifying even dimensions, and hence our consideration of the internal space is four dimensions, leading to a 6D SUSY field theory model.

The heterotic string theory has in the 4D viewpoint. To obtain chiral fermions, we have to reduce down to . The most famous internal space for this purpose is the Calabi-Yau space with holonomy[11]. The orbifold also reduces down to [4]. It can be understood by observing that blowing up the singularities of the orbifold fixed points leads to the Calabi-Yau space with the holonomy, because is the center of . To obtain a 6D model by compactifying four internal space, we cannot have an holonomy since 6D SUSY theory must be . But we find that orbifold works in reducing down to . It is because an vector 4 is under and giving a vacuum expectation value to 4 can break one but leaves the other unbroken, which leads to an holonomy. Then SUSY is reduced by half. Since is the center of , we obtain 6D (in 4D viewpoint) theory. To obtain an theory, one has to introduce another to break the remaining , which we do not consider in this paper. With the orbifold, we obtain the simplest 6D orbifold family unification model , which can be compared to the simplest 4D orbifold model [4]. It can lead to 5D flavor unification models considered in Refs.[9, 10]. Also, it is possible to consider a 6D model as a flavor unification model.

## Ii orbifold

In this spirit, let us proceed to consider the orbifold
compactification of with and in
mind.^{3}^{3}3Even though we are interested in and ,
some formulae include and explicitly to compare with the
well-known case and [4]. The twisting of the
-dimensional internal space and gauge groups are,

We define is the eigenvalue of the internal space rotation,

satisfying .

There are conditions to be satisfied[4].

(2) |

where for even(odd) . The first equation in Eq. (2) is from the definition of twist of order up to fermionic degree of freedom, and the second one is the modular invariance condition.

There are only few possibilities for the lattice vector and of satisfying these conditions. For should be again lattice vector in , is times an integer. Also it is known that any lattice vector in is within the distance 1 from some lattice point, so . There are only three allowed combinations which are shown in Table I.

Table I. orbifolds of the heterotic string. Case shift shift 6D gauge group (i)

Case (i) breaks down to . Cases (ii) and (iii) break down to . The cases with entries , and do not satisfy the two conditions given above. We will mainly discuss and comment on flavor unification group at the end.

### ii.1

Let us consider the first which can give 5D flavor unification models[9, 10]. Even though Cases (ii) and (iii) give the same theory, we will treat them separately, since the introduction of Wilson lines can be studied more easily with two different shift vectors. It is easy to understand Case (ii). It is simply separating 248 of into vector and spinor parts, where the vector forming the adjoint representation of and the spinor formining the matter fields 128 of [4]. Thus, we discuss Case (iii) only in detail since it has not been discussed in the literature and can be a potential GUF(grand unification of families) [9, 10]. The gauge group from can be considered as the “hidden” sector needed for SUSY breaking.

Now, from the mass shell condition

(3) |

we have massless states as follows:

On the left-moving side there are

(4) | |||||

(5) |

where we have not displayed the hidden sector. The corresponding lattice vectors are shown in the following tables.

Table II. Root vectors in untwisted sector transforming like . The underlined entries allow permutations and those in the [] bracket allow even numbers of sign flips. vector number of states

In the Tables, we have the convention that the underlined entries allow premutations and those in the square braket [] allow even numbers of sign flips. We have 112 winding states and from the oscillators we have 8 generators. They will form the adjoint representation 120 of .

Table III. Root vectors in untwisted sector transforming like . vector number of states

On the right moving side, there are vectors transforming like and ,

(6) | |||||

(7) |

where s are NS creation operators. Note that, the first entry of spinorial(Ramond) representation is chirality in six dimension. The (leading to gaugino after combining with the left movers) and (leading to the chiral matter after combining with the left movers) have opposite chiralities. For , a spinor of can be represented by -tuples of spin eigenstates as, where can be either . Then, the chirality is defined by the eigenvalue of

(8) |

Compactification of some internal space of even dimensions(e.g., from 10D to 6D) kicks out some factors of in Eq. (8) and the product of the remaining ’s in Eq. (8) determines the chirality in the uncompactified space. For example, the chirality in 6D in our compactification of 10D down to 6D is . The ten dimensional 8 of right-moving ground state is decomposed into two 2’s and four 1’s with opposite chirality in six dimension.

Combining the right movers of the preceeding paragraph with the left movers into invariant states, we have an adjoint representation 120 for the gauge multiplet and a spinor representation 128 for the hyper-multiplet. Defining the chirality of the gauge multiplet as left-handed, the chirality of the hyper-multiplet becomes right-handed. In addition, there appear the 6D supergravity multiplets, etc.

Notice, however, that combining the right-handed with
the left-handed 128 seems to give 2 spinors.^{4}^{4}4In
orbifold, this is the reason that we obtain three copies of chiral
fermions from the untwisted sector. But note that the center of
is for an even
and for an odd [13].
Our orbifolding under and assigning
at the center of still allow freedom. Thus, the two
spinors of are connected by gauge transformation,
and we have to divide the number of spinors by the left-over
center . Thus, the untwisted sector allows only
one spinor 128 of . It is similar to the mechanism
of calculating the domain wall number in axionic models[14].

### ii.2 Twisted sector

In the twisted sector, the twisted mode expansion gives a different zero point energy. The zero point energy of a bosonic string is given by is a shift. In our model, the zero point energy is , where

(9) |

where is the oscillator number. For , there is no vector in the lattice, satisfying Eq. (9). For , we have some lattice vectors satisfying the above condition. We can find two sixteen states satisfying Eq. (9) given in Table IV.

Table IV. Root vectors in the twisted sector vector number of states

Combining with the right-moving states, we have two 16’s in the twisted sector.

In the orbifold model, we encounter only fixed points. However in a general orbifold model with even, there are also fixed tori which have a different topology and can give a different number for the twisted sector states.

For the case of the orbifold, there are 16 fixed points. However,
the effective multiplicity is 8 rather than 16.
^{5}^{5}5This can be more transparent by reading off
the partition function.
In terms of the notation in Ref.[6],
projector operator is

Therefore, there appear sixteen()
copies of 16 from the twisted sector. These have the same
chirality as the matter representation from the untwisted
sector.^{6}^{6}6The vector for the right movers
in the notation of Ref.[6] can be

### ii.3 Anomaly

In 6D, there can exist a square anomaly. The anomaly of for is[15],

(10) |

with the normalization in units of the anomaly of the left-handed vector representation. It is important that in this model we have gaugino and matter fermion with opposite chirality. In six dimension, Weyl spinor is self-dual, or charge conjugate of one has the same chirality of itself.

We can check the anomaly cancellation with the fermion spectrum obtained in the preceeding section,

This again shows the consistency of having only one spinor representation due to the nontrivial center of after orbifolding with .

### ii.4

In 6D the square anomalies of 133(adjoint) and 56 of are absent. As for the anomaly cancellation, therefore, in 6D is like in 4D. As we studied the model previous subsection, we can repeat a similar analysis. The gauge group is or . The matter in the untwisted sector is listed in Table IV. In this subsection, we will comment the flavor unification briefly. In the untwisted sector, matter fields of Table IV arise. The states in Table IV constitute (56,2) of since the center of is . We divided the total number by 2, as we have done in the case. The flavor group is . Thus, we obtain the flavor doublet of 56. It is a 6D model.

Table IV. Root vectors in untwisted sector transforming like . vector number of states

In the twisted sector, we obtain following states satisfying ,

which constitute (56,1). Since there are 16 fixed points, there are 16 copies of (56,1). If we somehow remove the 56’s from the twisted sector, we can have a four generation model. Indeed, we can devise such a scheme by compactifying the 6th dimension by or by the Scherk-Schwarz mechanism. Since the number of components of spinors in 6D and 5D are the same, by the Scherk-Schwarz mechanism for example, we just distinguish the representation property. We assign the anti periodic boundary condition for all the wave functions in the compactification of the 6th dimension. The rotation in the 6th dimension is embedded in the group space so that the spinors of get an extra minus sign for the rotation. Therefore, sixteen (56,1)’s are projected out. In 5D we obtain an model with the matter (56,2).

The braching of 56 to representations is

(11) |

Thus, it is possible to pick up two 27’s in 4D by picking one 27 from the left-handed and one from the right-handed. Then, we obtain a four generation model since 56 is a flavor group doublet.

## Iii Conclusion

We considered the orbifold compactification of the heterotic string to obtain a 6D model with one spinor. We pointed out that the untwisted sector matter fields should be properly counted by considering the center of . One spinor can split into two spinors, and a 5D compactification leads to reasonable family unification models[10]. We also commented on the flavor unification.

###### Acknowledgements.

We thank Paul Frampton and Kyuwan Hwang for useful discussions. One of us(JEK) thanks Humboldt Foundation for the award. This work is supported in part by the BK21 program of Ministry of Education, and the KOSEF Sundo Grant.## References

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