Higgs doublets, split multiplets and heterotic SU(3)CxSU(2)LxU(1)Y spectra,” arXiv:hep-th/0409291

A methodology for computing the massless spectrum of heterotic vacua with Wilson lines is presented. This is applied to a specific class of vacua with holomorphic SU(5)-bundles over torus-fibered Calabi-Yau threefolds with fundamental group Z_2. These vacua lead to low energy theories with the standard model gauge group SU(3)_C x SU(2)_L x U(1)_Yand three families of quark/leptons. The massless spectrum is computed, including the multiplicity of Higgs doublets.

has structure group G × F , which spontaneously breaks E 8 to the gauge group X can be constructed as the quotient whereX is a simply connected Calabi-Yau threefold and F acts freely onX. V and V ′ coincide when pulled back toX and are denoted byṼ . The structure group ofṼ is G, while (4) and (5) become respectively. With respect to the subgroup G × H ⊂ E 8 , adṼ decomposes as where U i (Ṽ ) are the vector bundles associated with the irreducible representation U i of G and R i are the corresponding representations of H.
As discussed in [7], the massless spectrum is identified as where / D is the Dirac operator on X, ρ ′ (F ) specifies the F action on both H q (X, U i (Ṽ )) and R i and the superscript indicates the invariant part of the expression. Decomposing R i in terms of its irreducible F -representations A j , expression (11) becomes Here, B ij carries a representation of the gauge group S. Therefore, to compute the massless spectrum it suffices to determine the dimension of the space of F -invariants in H q (X, U i (Ṽ ))⊗ A j .
In this paper, we choose Then and (11) becomes To determine the massless spectrum, one must compute the cohomology groups in (16), the action of Z 2 on these groups and the action of Z 2 on each representation R i . Since the last of these is straightforward, we discuss it first.
For F = Z 2 , W spontaneously breaks H to the standard model gauge group The action of Z 2 on each representation R i of H is easily computed. For example, for R i = 5 expression (12) is where ±1 are the representations A j of Z 2 while (a, b) w are representations of S. For notational simplicity, we display w = 3Y . The action of Z 2 on each representation R i in (16), as well as the corresponding representations B ij of S, are listed in Table 1.
To compute the cohomology groups in (16), we must constructX andṼ . ChooseX to be the fiber productX of two dP 9 surfaces B and B ′ .X is elliptically fibered over both surfaces with the projections B and B ′ are themselves elliptically fibered over P 1 with the maps A Z 2 action τ onX can be obtained as the lift of two involutions τ B and τ B ′ on B and B ′ respectively. It is sufficient to know that τ B acts on P 1 as and SU(2) L respectively, whereas w = 3Y .
where t 0 , t 1 are projective coordinates. This action has two fixed points, p 0 and p ∞ . The has four fixed points.
The τ B ′ action on B ′ has similar properties. In order for τ in (22) to act freely onX, one must "twist" the two P 1 lines in (21) when identifying them in (19). This twist sets p ′ 0 = p ∞ and p ′ ∞ = p 0 . Stable, holomorphic vector bundlesṼ onX with structure group G = SU(5) can be constructed as the extension of two vector bundles with rkV i = i for i = 2, 3. W 2 and W 3 are vector bundles on B with rank 2 and 3 respectively, while L 2 and L 3 are line bundles on B ′ . We need to lift the Z 2 action onX to an action on V 2 , V 3 andṼ . ForṼ to be Z 2 invariant, it is necessary to restrict both V 2 and V 3 to be invariant. This is done by choosing W i and L i to be τ B and τ B ′ invariant respectively.
There are many line bundles L i that are invariant under τ B ′ . However, we now impose the remaining constraint thatṼ satisfy (9) with |F | = 2. This restricts the allowed line bundles to be where r ′ is a specific divisor of B ′ with deg(r ′ ) = 2 when restricted to a fiber. By construction, V corresponds to an extension class We can now construct the cohomology groups in (16). However, one group, H 1 (X, adṼ ), corresponding to vector bundle moduli, requires techniques beyond those developed in this paper and will not be discussed. Let us consider H 0 (X, OX). Since OX is the trivial bundle, it follows that Note that since OX is independent ofṼ , Z 2 acts trivially on H 0 (X, OX).
Next, we determine H 1 (X,Ṽ ). From the long exact sequence associated with (24), we find that Using (25) and pushing this down fromX to P 1 gives We find that R 1 β * W 2 ≃ O p∞ . From (26) it follows that β ′ * L 2 has degree 6 along f ′ 0 . We conclude that Now consider H 1 (X,Ṽ * ). This can be determined from (31) using the Atiyah-Singer index theorem which, together with Serre duality, gives where we have used (9) with |F | = 2. This and (31) then imply We now turn to the computation of H 1 (X, ∧ 2Ṽ ). One can show that H 1 (X, ∧ 2Ṽ ) lies in the exact sequence To continue, we must compute the terms H i (X, ∧ 2 V 2 ), i = 2, 3 and H 1 (X, V 2 ⊗ V 3 ), as well as the linear map M T . Pushing H i (X, ∧ 2 V 2 ), i = 2, 3 down to P 1 , we find and It follows that Calculating H 1 (X, V 2 ⊗ V 3 ) is more difficult. Using (25) and pushing down to P 1 , we find Here, we simply state that R 1 β * (W 2 ⊗ W 3 ) is a sheaf supported at each of 12 points p r ∈ P 1 , r = 1, . . . , 12 and at p ∞ . Specifically, Furthermore, it follows from (26) that β ′ * (L 2 ⊗ L 3 ) is a rank two vector bundle on P 1 . Combining these results, (38) becomes Finally, we must know the rank of M T . It follows from (37) and (40) Putting (37), (40) and (41) into (34), we conclude that Finally, we need to compute H 1 (X, ∧ 2Ṽ * ). Again, this is easily determined using the Atiyah-Singer index theorem. In this context, we find Combining this with (42) yields Having computed all the cohomology groups in (16), we now determine the explicit action of Z 2 on each of them. Let us begin with H 1 (X,Ṽ ), which was given in (31). To begin with, consider the second factor, C 6 . This can be shown to be parametrized by the polynomials where i = 0, . . . , 3 and j = 0, 1. Here, x 0 , x 1 and y are sections of specific bundles on the base P 1 , which transform as under τ B ′ . Applying these transformations to (45), we see that C 6 decomposes as C 3 (+) ⊕C 3

(−)
under the action of Z 2 . Since this is evenly split between + and −, the Z 2 action on the first factor C in (31) is irrelevant. We conclude that We now compute the Z 2 action on H 1 (X,Ṽ * ) in (33) using the Atiyah-Singer index theorem. First, consider the index theorem for V on X =X/Z 2 . Using (9) with |F | = 2, the fact that H q (X,Ṽ ) (+) = H q (X, V ) for any q and Serre duality, we find Using (9), Serre duality and (48), the index theorem forṼ onX becomes It then follows from (47), (48) and (49) that under the action of Z 2 .
Now consider H 1 (X, ∧ 2Ṽ ) in (42). It follows from (34) that to find the Z 2 action on H 1 (X, ∧ 2Ṽ ), one must determine its action on H i (X, ∧ 2 V 2 ), i = 1, 2 in (37), in (40) and on the map M T satisfying (41). Since the decomposition of each of these cohomology groups under Z 2 is computed using methods similar to those leading to (47), we simply state the results. We find and Furthermore, one can show that M T can be taken to be invariant under Z 2 , corresponding to choosing [Ṽ ] to be in Ext 1X (V 3 , V 2 ) (+) . Then, it follows from (51) and (52) that Putting (51) and (53) into the exact sequence (34), we conclude that Finally, we can compute the Z 2 action on H 1 (X, ∧ 2Ṽ * ) using the Atiyah-Singer index theorem. This computation is very similar to that leading to (50), so we will simply state the result. We find that We now possess all of the ingredients necessary to compute the massless spectrum. Combining (47), (50) and (54)-(55) with the results in Table 1, one can determine the ρ ′ (Z 2 ) invariant subspace for each cohomology group in (16). The associated multiplets descend to X =X/Z 2 to form the SU(3) C × SU(2) L × U(1) Y particle physics spectrum. The results are tabulated in Table 2.
To begin with, the spectrum contains one copy of vector supermultiplets transforming Furthermore, it contains three families of quarks and lepton superfields, each family transforming as respectively. However, there are additional chiral superfields in the spectrum. It follows from Table 2 that these occur as conjugate pairs of the SU (3) These multiplets arise as Z 2 invariants in the 5 and 10 representations of H = SU (5). The spectrum has n (3,1) −2 = 9, n (1,2) 3 = 9 (61) and n (3,1) 4 ⊕(1,1) −6 = 3, n (3,2) −1 = 3 copies of (59) and (60) respectively. The multiplicity n (1,2) 3 corresponds to the number of Higgs doublet conjugate pairs in the low energy spectrum. The remaining multiplets in (59) and (60) are exotic. Clearly, the number of Higgs doublets and the exotic multiplets is not consistent with phenomenology. However, we emphasize that these results were computed within a specific context, which is but a small subset of the possible standard model heterotic vacua. These generalized vacua and their spectra will be presented in future publications.