A Heterotic Standard Model

Within the context of the E_8 x E_8 heterotic superstring compactified on a smooth Calabi-Yau threefold with an SU(4) gauge instanton, we show the existence of simple, realistic N=1 supersymmetric vacua that are compatible with low energy particle physics. The observable sector of these vacua has gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each with an additional right-handed neutrino, two Higgs-Higgs conjugate pairs, a small number of uncharged moduli and no exotic matter. The hidden sector contains non-Abelian gauge fields and moduli. In the strong coupling case there is no exotic matter, whereas for weak coupling there are a small number of additional matter multiplets in the hidden sector. The construction exploits a mechanism for ``splitting'' multiplets. The minimal nature and rarity of these vacua suggest the possible theoretical and experimental relevance of spontaneously broken U(1)_{B-L} gauge symmetry and two Higgs-Higgs conjugate pairs. The U(1)_{B-L} symmetry helps to naturally suppress the rate of nucleon decay.

The structure of the hidden sector depends on whether one is in the weakly or strongly coupled regime of the heterotic string. In the strongly coupled context, we find the following minimal hidden sector.

2.
A small number of vector bundle moduli.

No matter fields.
Again, note that this hidden sector has no exotic matter. Combining this with the above, we have demonstrated, within the context of the strongly coupled heterotic string, the existence of realistic vacua containing no exotic matter fields. We emphasize that the hidden sector gauge group E 7 × U(6) is sufficiently large to allow acceptable supersymmetry breaking via condensation of its gauginos.
In the weakly coupled context, we find the following minimal hidden sector. (This is also a valid vacuum in the strongly coupled case).

A small number of vector bundle moduli.
3. Two matter field multiplets in the 12 of Spin (12).
Note that, in this case, there are a small number of exotic matter multiplets in the hidden sector. Again, the hidden sector gauge group Spin (12) is sufficiently large to allow acceptable supersymmetry breaking via gaugino condensation.
The vacua presented above are the result of an extensive search within the wide context made precise in [7]. They appear to be the minimal vacua, all others containing exotic matter fields, either in the observable sector, the hidden sector, or both, usually with a large number of Higgs-Higgs conjugate pairs. We have been unable to find any vacuum in this context with only a single pair of Higgs-Higgs conjugate fields. Furthermore, to our knowledge, phenomenological vacua in all other string contexts [6,8,10,11,12] have substantial amounts of exotic matter, both in the observable and hidden sectors. For all these reasons, we refer to the class of vacua presented in this paper as a heterotic standard model and speculate that it may be of phenomenological significance. In particular, it would seem to motivate renewed interest, both theoretical and experimental, in its characteristic properties; namely, 1) the physics of a U(1) B−L gauge symmetry spontaneously broken at, or above, the electroweak scale and 2) the physics of two pairs of Higgs-Higgs conjugate fields, particularly their experimental implications for flavor changing neutral currents. It is immediately clear that the B − L symmetry will help to naturally suppress the rate of nucleon decay. This potentially resolves a long-standing problem in phenomenological string vacua. At the least, our results go a long way toward demonstrating that realistic particle physics can be the low energy manifestation of the E 8 × E 8 heterotic superstring, as originally envisaged in [8,9].
We now specify, in more detail, the properties of the these minimal vacua and indicate how they are determined. Following [8], the requisite Calabi-Yau threefold, X, is constructed as follows. We begin by considering a simply connected Calabi-Yau threefold, X, which is an elliptic fibration over a rational elliptic surface, dP 9 . In a six-dimensional region of moduli space, such manifolds can be shown to admit a Z 3 × Z 3 group action which is fixed point free. It follows that is a smooth Calabi-Yau threefold that is torus-fibered over a singular dP 9 and has non-trivial fundamental group as desired. It was shown in [4] that X has Kähler and complex structure moduli respectively. To our knowledge, this is the only Calabi-Yau threefold with Z 3 × Z 3 fundamental group that has been constructed. We note [13] that the transpose of the configuration matrix [14] associated with X defines another simply connected Calabi-Yau threefold. Interestingly, this is precisely the manifold introduced by Tian and Yau [15] which, when quotiented by Z 3 , was used to construct three generation heterotic string vacua within the context of the standard gauge embedding.
We now construct a stable, holomorphic vector bundle, V , on X with structure group contained in the E 8 of the observable sector. This bundle admits a gauge connection satisfying the Hermitian Yang-Mills equations. The connection spontaneously breaks the observable sector E 8 gauge symmetry to as desired. We produce V by building stable, holomorphic vector bundles V with structure group SU(4) over X that are equivariant under the action of Z 3 ×Z 3 . This is accomplished by generalizing the method of "bundle extensions" introduced in [5]. The bundle V is then given as Realistic particle physics phenomenology imposes additional constraints on V . To ensure that there are three generations of quarks and leptons after quotienting out Z 3 × Z 3 one must require that where c 3 ( V ) is the third Chern class of V . Recall that with respect to SU(4) × Spin(10) the adjoint representation of E 8 decomposes as The number of 16 zero modes is given by h 1 ( X, V * ) [6]. Therefore, if we demand that there be no exotic matter fields arising from vector-like 16 − 16 pairs, V must be constrained so that Similarly, the number of 10 zero modes is h 1 ( X, ∧ 2 V ). However, since the Higgs fields arise from the decomposition of the 10, we must not set the associated cohomology to zero. Rather, we restrict V so that h 1 ( X, ∧ 2 V ) is minimal, but non-vanishing. Subject to all the constraints that V must satisfy, we find Finally, for the gauge connection to satisfy the Hermitian Yang-Mills equations the holomorphic bundle V must be stable. A complete proof of the stability of V is technically very involved and has not been carried out. However, there are a number of non-trivial checks of stability that can be made. Specifically, stability constrains the cohomology of V to satisfy We have shown [7] that vector bundles V satisfying the constraints eqs. (7), (9), (10) and (11) indeed exist. Henceforth, we will restrict our discussion to such bundles.
We now extend the observable sector bundle V by adding a Wilson line, W , with holonomy The associated gauge connection spontaneously breaks Spin (10) as where SU(3) C × SU(2) L × U(1) Y is the standard model gauge group. Since Z 3 × Z 3 is Abelian and rank(Spin(10)) = 5, an additional rank one factor must appear. For the chosen embedding of Z 3 × Z 3 , this is precisely the gauged B − L symmetry.
As discussed in [6], the zero mode spectrum of V ⊕ W on X is determined as follows. Let R be a representation of Spin (10), and denote the associated V bundle by U R ( V ). Find the representation The Atiyah-Singer index theorem, eq. (7) and (9) imply that We can calculate the Z 3 × Z 3 representation on H 1 ( X, V ) as well as the Wilson line action on 16.
Tensoring these together, we find that the invariant subspace consists of three families of quarks and leptons, each family transforming as and are the representations on 10 of the first and second generators. Tensoring these actions together, one finds that the invariant subspace consists of two copies of the vector-like pair That is, there are two Higgs-Higgs conjugate pairs occurring as zero modes of our vacuum.
Putting these results together, we conclude that the zero mode spectrum of the observable sector 1) has gauge group SU (3)  Thus far, we have discussed the vector bundle of the observable sector. However, the vacuum can contain a stable, holomorphic vector bundle, V ′ , on X whose structure group is in the E ′ 8 of the hidden sector. As above, one can construct V ′ by building stable, holomorphic vector bundles V ′ over X which are equivariant under Z 3 × Z 3 using the method of "bundle extensions". V ′ is then obtained by taking the quotient of V ′ with Z 3 × Z 3 . The requirement of anomaly cancellation relates the observable and hidden sector bundles, imposing the constraint that must be an effective class. Here c 2 is the second Chern class. In the strongly coupled heterotic string, [W] is the class of the holomorphic curve around which a bulk space five-brane is wrapped.
In the weakly coupled case [W] must vanish. We have previously constructed X and V and, hence, can compute c 2 (T X) and c 2 ( V ). Then eq. (22) becomes a constraint on the hidden sector bundle V ′ . The easiest possibility is that V ′ is the trivial bundle. However, in this case, we find that [W] is not effective.
The next simplest choice is to take V ′ to have structure group in E ′ 8 . This spontaneously breaks the hidden sector E ′ 8 symmetry to Recall that with respect to SU(2) × E 7 the adjoint representation of E ′ 8 decomposes as We now require that there be no exotic matter fields in the hidden sector. This imposes the additional constraint that Finally, the requirement that V ′ be stable implies the conditions  [16], we find that the five-brane gauge group is Moving in the moduli space of the holomorphic curve, this group can be maximally broken to U(1) 6 . We conclude that, within the context of the strongly coupled heterotic string, our observable sector is consistent with a hidden sector with gauge group E 7 × U(6) and no exotic matter. There are, additionally, a small number of uncharged vector bundle moduli that arise from the invariant subspace of H 1 ( X, V ′ ⊗ V ′ * ) under Z 3 × Z 3 , as well as some five-brane moduli.
We now exhibit a hidden sector, compatible with our observable sector, that has no five-branes; that is, for which This does not occur for structure group G ′ = SU (2). From the results in [17], we expect that the appropriate group may be the product of two non-Abelian groups. The simplest choice is This bundle, which is the sum of two SU(2) factors V ′ = V ′ 1 ⊕ V ′ 2 , spontaneously breaks the hidden sector E ′ 8 gauge group to E ′ 8 −→ Spin (12) .
With respect to SU(2) × SU(2) × Spin (12) The hidden sector will have no exotic matter fields if and Finally, note that the stability of each bundle V ′ i , i = 1, 2 implies the conditions Subject to eq. (22) and the condition eq. (29) that there be no five-brane, we are unable to simultaneously satisfy all of the constraints in eqs. (33), (34) and (35). Demanding that the stability conditions eq. (35) hold, it is possible to choose V ′ i , i = 1, 2 so that only the first condition in eq. (33) is fulfilled. One finds that, minimally, and However, we can show that the Z 3 × Z 3 action on H 1 ( X, V ′ 2 ) has no invariant subspace. It follows that the associated matter fields will be projected out under the quotient by Z 3 ×Z 3 . Unfortunately, this is not the case for H 1 ( X, V ′ 1 ⊗ V ′ 2 ). Here, we find that the Z 3 × Z 3 action is two copies of its regular representation, which leaves a two-dimensional subspace of H 1 ( X, V ′ 1 ⊗ V ′ 2 ) invariant. Hence, after quotienting by Z 3 × Z 3 , one finds two 12 multiplets of Spin (12). We conclude that, for vacua with no five-branes, our observable sector is consistent with a hidden sector with gauge group Spin(12) and two 12 multiplets of exotic matter. There are also vector bundle moduli arising from the Z 3 × Z 3 invariant subspace of H 1 ( X, V ′ ⊗ V ′ * ). These vacua can occur in the context of both the weakly and strongly coupled heterotic string.