#### SO(32) heterotic line bundle models

HJE
SO(32) heterotic line bundle models
Hajime Otsuka 0
0 Department of Physics, Waseda University
We search for the three-generation standard-like and/or Pati-Salam models from the SO(32) heterotic string theory on smooth, quotient complete intersection CalabiYau threefolds with multiple line bundles, each with structure group U(1). These models are S- and T-dual to intersecting D-brane models in type IIA string theory. We find that the stable line bundles and Wilson lines lead to the standard model gauge group with an extra U(1)B−L via a Pati-Salam-like symmetry and the obtained spectrum consists of three chiral generations of quarks and leptons, and vector-like particles. Green-Schwarz anomalous U(1) symmetries control not only the Yukawa couplings of the quarks and leptons but also the higher-dimensional operators causing the proton decay.
Superstrings and Heterotic Strings; Superstring Vacua
Consistency conditions in the low-energy effective action
Matter content and group decomposition
U(1)Y masslessness conditions
Pati-Salam-like models
3
Three-generation models on quotient complete intersection CY
mani
Standard-like models via the Pati-Salam-like symmetry
Explicit three-generation models
4
Conclusion
A Net-number of chiral fields B List of CICYs to be applicable for the Pati-Salam models
1 Introduction 2 Setup
2.1
2.2
2.3
2.4
folds
3.1
3.2
1
Introduction
String theory is a most successful candidate of a unified theory including both the gauge
and gravitational interactions. Among the perturbative superstring theories, the heterotic
string theory [1, 2] is an attractive one, since the gauge group and matter representations are
uniquely determined by the ten-dimensional gauge and gravitational anomaly cancellation
conditions. In the low-energy effective action of the heterotic string, the background gauge
field strength is related to the curvature of internal manifold through the Bianchi identity.
The simplest approach to solve the Bianchi identity is the “standard embedding”, where
the gauge bundle with SU(
3
) structure group is identified with the holomorphic tangent
bundle of the Calabi-Yau (CY) manifolds [3]. (See, e.g. refs. [4, 5] for the detailed analysis of
three-generation models.) On the other hand, it is possible to construct consistent heterotic
models on smooth CY threefolds, even if the gauge bundles are not directly related to the
tangent bundle [6]. In this approach, holomorphic gauge bundles satisfying the Hermitian
Yang-Mills equations lead to semi-realistic standard-like models as discussed in several CY
threefolds with stable non-abelian vector bundles [7–12] and line bundles [13, 14].
So far, E8 × E8 heterotic string models have been well discussed in contrast to SO(32)
heterotic ones. This is because the adjoint representation of E8 gauge group naturally
involves the spectrum of E6, SO(10) and SU(
5
) grand unified theories (GUTs). On the
– 1 –
are constructed on toroidal orbifold [16–19], torus with magnetic fluxes [20, 21], elliptically
fibered CY manifolds with stable vector bundles [22] given by the spectral cover
construction [23, 24] and smooth CY threefolds with line bundles [25]. However, the SO(32)
heterotic line bundle models on smooth CY threefolds have not been fully explored.
S- and T-dualities tell us that the SO(32) heterotic line bundle models correspond
to intersecting D6-brane models in type IIA string theory, where several stacks of branes
directly lead to the minimal supersymmetric standard model (MSSM) and/or Pati-Salam
model [26–28].1 It therefore motivates us to search for standard-like models from the
SO(32) heterotic string theory on CY threefolds without an intermediate GUT.2 To obtain
realistic three-generation models, we consider multiple line bundles in the Cartan directions
of SO(32) rather than non-abelian bundles. These line bundles allow us to decompose the
SO(32) gauge group and compute the net chiral asymmetries of quarks and leptons, taking
into account that the choice of line bundles is constrained by the consistency conditions
such as anomaly cancellation conditions, masslessness conditions of the hypercharge gauge
boson, supersymmetric conditions, and so-called K-theory condition. It turns out that these
theoretical and phenomenological requirements indicate the enhancement of the standard
model gauge group to the Pati-Salam one. To break the Pati-Salam gauge symmetry, we
concentrate on quotient complete intersection Calabi-Yau manifolds (CICYs) [30, 31] by a
similar argument as in E8×E8 heterotic line bundle models [13, 14], where one can introduce
Wilson lines into the internal components of U(
1
)s because of the existence of a
freelyacting discrete symmetry group.
We systematically search for standard-like models on
several CICYs with multiple line bundles, where the number of K¨ahler moduli is restricted
to 1 ≤ h1,1
≤ 5, with h1,1 being the hodge number of CICYs. We find that such restrictive
line bundles and certain Wilson lines lead to the standard-like models where the gauge
group consists of SU(
3
)C × SU(
2
)L × U(
1
)Y × U(
1
)B−L via the Pati-Salam-like symmetry.
The spectrum in the visible sector contains the three generations of quarks and leptons
without chiral exotics, and Higgs doublets appear as the vector-like particles with respect to
the standard model gauge group with an extra U(
1
)B−L. The gauge symmetries of the
lowenergy effective action allow for the perturbative Yukawa couplings of quarks and leptons.
The remainder of this paper is organized as follows. We first briefly review several
consistency conditions on the basis of the low-energy effective action of the heterotic string
theory in section 2.1. In section 2.2, we present the group decomposition of SO(32) gauge
symmetry employing multiple line bundles along the line of ref. [20]. Even if line
bundles satisfy the consistency conditions, they are further constrained by the masslessness
conditions of the U(
1
)Y gauge boson, originating from the couplings between the U(
1
)Y
gauge boson and closed string axions as discussed in section 2.3. A search for concrete
three-generation standard-like models on several quotient CICYs indicates in section 3
1For more details, see e.g. ref. [29] and references therein.
2For the model building realizing the SU(
5
)-like spectrum, we refer to ref. [25].
– 2 –
HJEP05(218)4
that the visible gauge group consists of SU(
3
)C × SU(
2
)L × U(
1
)Y × U(
1
)B−L as a result of
multiple line bundles and certain Wilson lines. Other U(
1
) gauge bosons become massive
through the Green-Schwarz mechanism. The obtained spectrum contains the three
generations of quarks and leptons, vector-like Higgs and extra vector-like particles with respect
to SU(
3
)C × SU(
2
)L × U(
1
)Y × U(
1
)B−L. Extra U(
1
) gauge symmetries including U(
1
)B−L
control the Yukawa couplings among the elementary particles and higher-dimensional
operators causing the proton decay. Finally, section 4 is devoted to the conclusion.
Consistency conditions in the low-energy effective action
We briefly review the low-energy effective action of the SO(32) heterotic string theory on
CY manifolds with multiple line bundles. (For more details, we refer to refs. [10, 11, 32].)
At the order of α′, the bosonic part of the low-energy effective action is given by
HJEP05(218)4
Sbos =
1 Z
where φ10 denotes the dilaton. Trace of F and R, “tr”, is taken in the fundamental
representations of SO(32) and SO(
1, 9
), κ10 and g10 are the gravitational and gauge couplings
normalized as 2κ210 = (2π)7(α′)4 and g10 = 2(2π)7(α′)3.
2
Zumino term in the presence of heterotic five-branes with tension T5 = ((2π)5(α′)3)−1.
Heterotic (anti) five-branes wrapping the holomorphic two-cycles γs correspond to the
′
positive (negative) Ns and the Poincar´e dual four-form of γs is represented as δ(γs).
H = dB(
2
)
− α4 (wYM − wL) involves the gauge and gravitational Chern-Simons
threeforms, wYM and wL. Note that Kalb-Ramond two-form B(
2
) is related to B(6) under the
Here, we include the Wess
ten-dimensional Hodge duality, namely ∗dB(
2
) = e2φ10 dB(6).
Throughout this paper, we consider the following internal gauge bundle (2.2) (2.3)
where La are the multiple line bundles, each with structure group U(
1
). The concrete
embedding of U(
1
) into SO(32) is discussed later. The inclusion of such line bundles
breaks SO(32) into the four-dimensional gauge group G and the adjoint representation of
SO(32) is decomposed as
496 →
M(Rp, Cp),
p
where Rp and Cp stand for certain representations of G and W . Given these line bundles, we
can construct standard-like models only if internal gauge bundles satisfy several consistency
conditions which are enumerated as follows.
W = M La,
M
a=1
– 3 –
First of all, when we denote the internal background field strengths as F¯ and R¯, the
Bianchi identity of the Kalb-Ramond field B(
6
)
d(e2φ10 ∗ dB(
6
)) = − 4
α
′
trF¯2
− trR¯2 − 4(2π)2 X Nsδ(γs)
s
!
constrains the abelian bundles as
ch2(W ) + c2(T M) = X Nsδ(γs),
s
where ch2(W ) and c2(T M) are the second Chern character and second Chern class of W
and the tangent bundle of the CY manifold M respectively. To keep the stability of the
system, we require the vanishing anti heterotic five-branes, namely
HJEP05(218)4
ch2(W ) + c2(T M) ≥ [0],
in cohomology.
Next, since the spinorial representation appears in the first excited mode in the
heterotic string [1, 2], we require that the first Chern class of the total gauge bundle lies in
second even integral cohomology basis of the CY manifold:
c1(W ) = X nac1(La) ∈ H2(M, 2Z),
a
Fij = F¯¯i¯j = 0,
¯
gi¯jF¯i¯j = 0,
where na are the integers depending on the embedding of line bundles into SO(32). Such a
condition is also related to the K-theory condition [33, 34] in the S-dual type I superstring
theory.
Finally, the nonvanishing U(
1
) field strengths on the CY manifold are constrained by
the supersymmetric condition. From the supersymmetric transformations of the gauginos,
the internal field strengths have to obey
implying that the vector bundles should be holomorphic. It is known that the constraint
equation for the (
1, 1
) form of F is called the Hermitian Yang-Mills equation. It can be
solved when holomorphic bundles satisfy the following condition
μ(La) =
1 Z
2ls4
M
J ∧ J ∧ c1(La) +
V Z
6πs
M
c31(La) +
1
4 c1(La) ∧ c2(T M) = 0,
for all a. J = ls2 Pih=1,11 tiwi is the K¨ahler form expressed in terms of a basis wi of H1,1(M, R),
where ti denote the K¨ahler moduli normalized by the string length ls = 2π α . Now,
√ ′
we include the string loop corrections [10, 11] characterized by the volume of CY V =
16 dijktitj tk and the dilaton s = V/(2πgs2). dijk stand for the triple intersection numbers
of CY manifolds, namely dijk = RM wi ∧ wj ∧ wk. Hence, the first Chern class of each
line bundle should be properly chosen such that the moduli spaces of K¨ahler moduli reside
in the supergravity reliable domain ti > 1 in string units. Otherwise, the nonvanishing
– 4 –
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
D-terms appear in the four-dimensional supergravity action. In this paper, we restrict
ourselves to line bundles satisfying all the consistency conditions presented so far.
In this approach, we can calculate the net-number of chiral massless fermions. When we
denote the internal bundle Cp to each Cp, the left-handed and the right-handed fermionic
zero-modes in Rp are counted by the Dolbeault cohomology H1(M, Cp) and H2(M, Cp)
respectively. Since the μ-stable bundles (μ(La) = 0) give the vanishing zeroth and third
Hirzebruch-Riemann-Roch theorem tells us that the net-number of chiral massless fermions
(chiral supermultiplets) is counted by the corresponding Euler number
eq. (2.2) are trivial bundles OM, the U(
1
) gauge symmetries are enhanced to non-abelian
ones as shown in section 3 and corresponding cohomology becomes dim(H0(M, OM)) =
dim(H3(M, OM)) = 1 and dim(H1(M, OM)) = dim(H2(M, OM)) = 0. This is because
the zero-modes of the Dirac operator are the (0, 0) and (0, 3) forms under the Dolbeault
operator ∂¯ on manifolds of SU(
3
) holonomy. For more details we refer to ref. [35].
In this way, proper internal line bundles have the potential to yield three generations
of chiral fermions for a large class of CY manifolds, although, in the standard embedding
scenario, three-generation models are restricted to specific CY manifolds with small hodge
numbers.
2.2
Matter content and group decomposition
In the following, we proceed to extract standard-like models from the SO(32) heterotic
string theory. Along the line of ref. [20], we first decompose the SO(32) gauge group as
SO(32) → SO(12) × SO(20),
496 → (
1, 190
) + (12v, 20v) + (
66, 1
),
where SO(12) and SO(20) are further decomposed by the insertion of line bundles,
SO(12) → SO(8) × SU(
2
)L × U(
1
)1 → SU(
4
)C × U(
1
)2 × SU(
2
)L × U(
1
)1
→ SU(3)C × U(
1
)3 × U(
1
)2 × SU(
2
)L × U(
1
)1,
SO(20) → U(
1
)4 × · · · × U(
1
)13.
3As commented in ref. [14], non-trivial line bundles La have dim(H0(M, La)) = 0 if μ(La) < 0 and also
dim(H3(M, La)) = 0 if μ(La) > 0. Such positive and negative slopes μ(La) exist in the K¨ahler moduli space,
since we focus on the case with μ(La) = 0. Hence, we obtain dim(H0(M, La)) = dim(H3(M, La)) = 0
according to the theorem in ref. [36].
– 5 –
(2.10)
HJEP05(218)4
(2.11)
(2.12)
(
15, 1
)0,0
66
Under the above decomposition, we take the Cartan directions of SO(32), Ha(a =
1, 2, · · · , 16), as H1 − H2 and H1 + H2 − 2H3 for SU(
3
)C and H5 − H6 for SU(
2
)L. Other
U(1) directions are chosen as
U(
1
)1 : (0, 0, 0, 0, 1, 1; 0, · · · , 0),
U(
1
)2 : (1, 1, 1, 1, 0, 0; 0, · · · , 0),
U(
1
)3 : (1, 1, 1, −3, 0, 0; 0, · · · , 0),
U(
1
)4 : (0, 0, 0, 0, 0, 0; 1, 0, · · · , 0),
U(
1
)5 : (0, 0, 0, 0, 0, 0; 0, 1, 0, · · · , 0),
U(
1
)13 : (0, 0, 0, 0, 0, 0; 0, · · · , 0, 1),
(2.13)
in the basis Ha and SO(32) roots are (±1, ±1, 0, · · · , 0) under Ha, (a = 1, · · · , 16). The
underline represents all the possible permutations.
Under this Cartan basis, the adjoint representation of SO(12), 66, involves all the
candidates of the standard model particles, except for the right-handed leptons as summarized
in table 1. Hence, it motivates us to further decompose SO(20) into multiple U(
1
)s. When
SO(20) is decomposed into all the U(
1
)s, the candidates of right-handed leptons appear
– 6 –
from the vector representation and the singlet of SO(12), 12v and 1,
cR2a = (3¯, 1)0,−1,−1;−1(a) , (a = 4, 5, · · · , 13),
, (a, b = 4, 5, · · · , 13, a < b),
(2.14)
where the subscript indices label the U(
1
)1,2,3 and non-zero U(
1
)a,b charges. This
decomposition results in the candidates of right-handed quarks, charged-leptons and/or Higgs.
To realize the correct hypercharge, we redefine the hypercharge as 1 6
13
c=4
!
U(
1
)Y =
U(
1
)3 + 3 X U(
1
)c ,
(2.15)
and the matter content and associated cohomology are then summarized in table 2. It
is notable that we just decompose SO(20) into tenth U(
1
)s by inserting line bundles into
all the Cartan directions. If the first Chern numbers of certain U(
1
)s are vanishing or
correlated with each other, these U(
1
) gauge symmetries are enhanced to non-abelian ones
as demonstrated in section 2.4.
2.3
U(
1
)Y masslessness conditions
Before searching for three generations of quarks and leptons, let us discuss the couplings
between string axions and the U(
1
)Y gauge boson. Such axionic couplings will cause the
mass term of the U(
1
)Y gauge boson, even when gauge bundles satisfy the consistency
conditions presented in section 2.1. The authors of refs. [10, 11] pointed out that background
gauge fluxes induce the couplings between string axions and U(
1
) gauge bosons through
the Green-Schwarz term [37, 38],
where
X8 =
TrF 4
1
24
1
Under the expansion of the Kalb-Ramond field in a basis of H2(M, Z),
SGS =
1
24(2π)5α′
Z
B(
2
)
∧ X8,
B(
2
) = b(02) + ls2 X bi(0)wi,
B(6) = ls6b(00)vol6 + ls4 X bi(
2
)wˆi,
h1,1
i=1
h1,1
i=1
– 7 –
(2.16)
(2.17)
(2.18)
Q1
Q2
L1
L2
La
3
La
4
c
uR1
ucR2a
c
dR1
c
dR2
dcR3a
ecR1a
ecR2ab
n1
nc a
2
nc ab
3
Repr.
indices label the U(
1
)1,2,3 and non-zero U(
1
)a,b charges.
dimensional hodge duality ∗dB(
2
) = e2φ10 dB(6).
where wˆi are the Hodge dual four-forms of the K¨ahler forms wi and vol6 denotes the volume
form of the CY manifold, bi(0) and b(00) represent the model-dependent and -independent
axions respectively. Those are connected with two-forms, bi(
2
) and b(02) under the
ten
On the line bundle background, the U(1)a gauge field strengths are decomposed as
(2.19)
(2.20)
where fa are four-dimensional parts and
denote the background gauge fluxes in a basis of H2(M, Z). Here, mia are the integers
subject to the Dirac quantization condition and we assume that the first Chern numbers
of all the U(
1
)s are independent, otherwise these U(
1
) gauge symmetries are enhanced to
non-abelian ones. Such a gauge background gives rise to the axionic couplings through the
Fa = fa + f¯a,
f¯a = 2π X miawi
h1,1
i=1
– 8 –
Green-Schwarz term,
1
3(2π)3ls2
Z b(
2
)
for the model-independent axion4 and
0 ∧ trT14f¯13f1 + trT24f¯23 + 3(trT22T32)f¯2f¯32 + (trT2T33)f¯33 f2
(2.21)
+ trT34f¯33 + 3(trT2T33)f¯2f¯32 + 3(trT22T32)f¯22f¯3 f3 + X trTc4f¯c3fc ,
13
c=4
l
2
s
1 Z b(
2
)
i ∧
13
a=1
X tr(Ta2)mia
!
fa,
(2.22)
for the model-dependent axions.5 Here, Ta denote the U(
1
)a generators whose directions
are chosen as in eq. (2.13). These Stueckelberg couplings yield the mass terms of the U(
1
)a
gauge bosons as mentioned before.
3 Pc1=34 U(
1
)c), the masslessness of the U(
1
)Y gauge field is ensured when
Recalling the fact that the hypercharge U(
1
)Y is defined as U(
1
)Y = 16 (U(
1
)3 +
tr(T34)dijkmi3mj3m3k + 3tr(T2T33)dijkmi2mj3m3k + 3tr(T22T32)dijkmi2mj2m3k
13
c=4
+3 X tr(Tc4)dijkmicmjcmck = 0
and
are satisfied simultaneously.
2.4
Pati-Salam-like models
The simplest way to satisfy all the requirements is to set
tr(T32)mi3 + 3 X tr(Tc2)mic = 0
13
c=4
mi5 = −m4,
i
mi3 = mid = 0,
for d = 6, · · · , 13 and i = 1, 2, · · · , h1,1 and non-zero values for other line bundles. Under
the above ansatz, one can easily satisfy the K-theory condition
c1(W ) = 2c1(L1) + 4c1(L2) + X c1(Lc) = 0 (mod 2),
and the U(
1
)Y masslessness conditions in eqs. (2.23) and (2.24). Here and in what follows,
the traces of U(
1
) generators are taken as tr(T1) = 2, tr(T2) = 4, tr(Tc) = 1, where
U(
1
)1 and U(
1
)2 are embedded into U(
2
)L and U(4)C gauge groups respectively. Note
4Note that the curvature terms in eq. (2.17) are irrelevant to the masslessness conditions of U(
1
)Y .
5For the detailed derivation, see ref. [20].
– 9 –
13
c=4
(2.23)
(2.24)
(2.25)
(2.26)
that the gauge symmetries SU(
2
)L and SU(4)C of the Pati-Salam model are embedded as
SU(
2
)L ⊂ U(
2
)L and SU(
4
)C ⊂ U(
4
)C . These first Chern classes are of the form
c1(La) = X miawi,
h1,1
i=1
(2.27)
where the quantized fluxes mia are the integers as mentioned before. However, the above
choice of line bundles results in the Pati-Salam-like gauge group plus hidden gauge group,
SO(32) → SU(
4
)C × SU(
2
)L × SU(
2
)R × Πc=1U(
1
)c × U(
1
)′ × SO(16),
2
(2.28)
where SU(
3
)C is enhanced to SU(
4
)C due to the vanishing c1(L3). The correlated first
Chern classes mi5 = −mi4 also lead to the gauge enhancement from U(
1
)4,5 to SU(
2
)R⊗U(
1
)′
whose Cartan directions are taken as H4+H5 for SU(
2
)R and H4−H5 for U(
1
)′ ≃ S(U(
1
)4×
U(
1
)5). For our purpose, we search for the three-generation models using S(U(
1
)4 × U(
1
)5)
rather than U(
1
)′ in the latter analysis, where two U(
1
)4,5 charge vectors q = (q4, q5)
and q′ = (q4′, q5′) are identified if q − q′ ∈ Z × (
1, 1
) because of L5 ≃ L4−1. Other U(
1
)s,
especially SO(
2
)s, are enhanced to hidden SO(16) due to the vanishing first Chern classes.
As a result, the remaining gauge symmetry on M is similar to the Pati-Salam model. It
is remarkable that the above discussion presented so far is irrelevant to the underlying CY
geometries thanks to the constraints on the line bundles (2.25). Although it is possible to
directly derive the standard model gauge group with hypercharge flux breaking, we require a
concrete model-by-model search. We therefore leave the detailed analysis for a future work.
When the vacuum expectation values of singlet fields under the Pati-Salam-like gauge
group are taken zero, the number of massive U(
1
)s is given by the rank of the following
mass matrix in units of string length ls = 2π
√
djkl tr(Ta) 16 mjamkamla + 214 mjac2kl(T M)
for i = 0
.
(2.29)
The first line is derived from the Stueckelberg couplings between the model-dependent
axions and U(
1
)a gauge bosons, whereas the second line is originating from the
modelindependent axion associated with the dilaton field. It turns out that when h1,1 = 1,
the maximum rank of mass matrix (2.29) is 2 in which at least one of the three U(
1
)s
remains massless due to the correlated Chern numbers (2.25). When h1,1
≥ 2, it is possible
to consider three massive U(
1
) gauge bosons at the compactification scale.
Note that
the Cartan direction of U(
1
)Y = 61 (U(
1
)3 + 3U(
1
)4 + 3U(
1
)5) remains massless and the
remaining U(
1
)3 could be identified with U(
1
)B−L after the Pati-Salam symmetry breaking.
In addition to the constraints on the line bundles (2.25), we restrict ourselves to line
bundles satisfying the D-term conditions (2.9) and the stability condition (2.6) simplified as
2ch2(L1) + 4ch2(L2) + ch2(L4) + ch2(L5) + c2(T M) ≥ [0],
(2.30)
in cohomology, where ch2(La) = 12 c21(La) = 12 dijkmjamkaw˜i with ci1(La) = mia. Note that
we use the relations tr(T12) = 2, tr(T22) = 4 and tr(T42,5) = 1. In the light of D-term
conditions (2.9), it is difficult to realize the vanishing D-terms for h1,1 = 1. Hence, we
demand h1,1
≥ 2, yielding three massive U(
1
) gauge bosons except for U(
1
)Y and U(
1
)B−L.
One option to break the Pati-Salam symmetry is its spontaneous symmetry breaking
in the presence of vector-like Higgs (4¯, 1, 2, 1) + c.c.. In this paper, we take into account
another option, namely to introduce Wilson lines into the internal components of SU(
4
)C
and SU(
2
)R on the quotient CY manifolds.
3
Three-generation models on quotient complete intersection CY
manifolds
We are now ready to searching for the three-generation models on CY manifolds with
multiple line bundles satisfying all the consistency conditions. In particular, we concentrate
on smooth, quotient CICYs M˜ =
M/Γ [30, 31], where M is a simply connected CICY
with a freely-acting discrete symmetry group Γ. CICYs defined in the ambient space
Pn1 × · · · ×
Pnm are characterized by the following m × R configuration matrix,
HJEP05(218)4
Pn1 q11 q2 · · · qR1
1
... ...
. ...
where the subscript and superscripts denote the Euler number and Hodge numbers of
CICYs respectively.
When the homogeneous coordinates of Pnl are represented as xl
multi-degree of R homogeneous polynomials on Pn1 × · · · ×
with l = 1, · · · , m and α = 0, · · · , nl, the positive integers qrl (r = 1, · · · , R) specify the
Pnm . Then, the common zero
locus of such R polynomials corresponds to the defining equations of CICYs. The CY
α
condition, namely a vanishing first Chern class of the tangent bundle, is realized by setting
PrR=1 qrl = nl + 1, for all l and the dimension of CICYs becomes 3 under Plm=1 nl − R = 3.
Through out this paper, we focus on the favorable CICYs where the second cohomology of
CY descends from that of the ambient space.
Among them, some of CICYs admit a freely-acting discrete symmetry group Γ classified
in ref. [39]. Those quotient CICYs are of particular importance, since one can turn on
Wilson lines because of the existence of the discrete symmetry group. Furthermore, Γ
reduces the number of complex structure moduli.6 As discussed in section 2.4, some of the
line bundles satisfying all the conditions in section 2 should be correlated with one another.
The gauge symmetry of the standard model in turn is enhanced to the Pati-Salam-like
symmetry. We therefore introduce Wilson lines into the internal components of U(
1
)s on
the quotient CICYs such that the Pati-Salam group is broken to the standard model one.
It turns out that one cannot find realistic models on CICYs with h1,1 = 1, 2, 3, 5, since the
underlying CY geometries and consistency conditions constrain the choice of background
6In general, the number of K¨ahler moduli on M and M˜ = M/Γ is different from each other, but in this
paper we focus on CICYs with h1,1(M) = h1,1(M˜ ).
line bundles. In the next but one subsection 3.2, we show the standard-like models on a
concrete quotient CICY with h1,1(M˜ ) = 4.
Standard-like models via the Pati-Salam-like symmetry
To break the Pati-Salam-like symmetry into the standard model one, we consider the
quotient CICYs M˜ = M/Γ, where one can introduce Wilson lines into the internal
components of U(
1
)3,4,5 such that SU(4)C and SU(
2
)R × U(
1
)′ are broken to SU(
3
)C × U(
1
)3
and U(
1
)4,5 respectively. (For more details, we refer to ref. [35].)
When the discrete
symmetry group Γ consists of only Abelian discrete symmetries Zn, namely Γ = ⊗iZni , the
representations of Z
ni are represented as e2πipi/ni for pi ∈ {0, 1, · · · , ni − 1}. If the Wilson
lines are in the center of SU(
4
)C and SU(
2
)R, the Pati-Salam symmetry is unbroken.
Hence, one of the Abelian symmetries should be different from Z4 and Z2. In such a case,
the low-energy gauge group on the quotient CICYs M˜ becomes
5
SO(32) → SU(3)C × SU(
2
)L × Πc=1U(
1
)c × SO(16),
(3.2)
where three U(
1
)s except for U(
1
)Y and U(
1
)B−L become massive through the axionic
couplings.
On quotient CICYs M˜ , the net-number of massless chiral supermultiplets is subject
to the discrete symmetry group Γ. Indeed, the gauge bundles on M reduce to those on M˜
if such bundles are equivariant bundles, and the cohomology associated with the matter
field on M˜ is described by the subspace of H1(M, V ). Note that all line bundles have
an equivariant structure for a single Zn, namely Γ = Zn. (For more details, we refer to
ref. [14].) The number of generation on M is divided by the group order, |Γ|:
χ(M, Cp) ,
Γ
| |
where S is a certain representation of Γ. Note that the chiral asymmetry is independent
of Wilson lines.
that the indices in table 3 become
Under these circumstances, we search for three generations of quarks and leptons such
m(
4,2,1,1
)1,1,0 + m(
4,2,1,1
)−1,1,0 = χ(M, L1 ⊗ L2) + χ(M, L1−1 ⊗ L2) = −3|Γ|,
m(¯4,1,2,1)0,−1,−1 + m(¯4,1,2,1)0,−1,1 = χ(M, L2−1 ⊗ L4−1) + χ(M, L2−1 ⊗ L4) = −3|Γ|,
within −3|Γ| ≤ m(
4,2,1,1
)±1,1,0 , m(¯4,1,2,1)0,−1,±1 ≤ 0 and indices of other massless states in
table 3 are integrally quantized on M and M˜ . Now, we have used c1(L3) = 0 and each index
is calculated using the formula in appendix A. Note that we have not taken into account
the generation of Higgs and Higgsino fields which could be identified with L43,,54 and/or
their conjugates in the latter analysis. We further require that the exotic modes such as
(
4, 1, 1, 16
)0,−1,0, (
1, 2, 1, 16
)1,0,0 and (
1, 1, 2, 16
)0,0,1 in table 3 should not be chiral, namely
(3.3)
(3.4)
χ(L1) = 0,
χ(L2) = 0,
χ(L4) = −χ(L5) = 0.
(3.5)
When the heterotic five-branes present in the system, there exist chiral fermions under
the fundamental representations of SO(32) × Sp(2Ns), namely (32, 2Ns). Here, Ns denotes
the number of heterotic five-branes which have the symplectic gauge degrees of freedom
as confirmed in the S-dual type I superstring theory [40]. Since these chiral fermions
have (4, 1, 1, 16, 2Ns)0,−1,0, (1, 2, 1, 16, 2Ns)1,0,0 and (1, 1, 2, 16, 2Ns)0,0,1 representations of
SU(
4
)C × SU(
2
)L × SU(
2
)R × Πc2=1U(
1
)c × U(
1
)′ × SO(16) × Sp(2Ns), the condition (3.5)
also ensures the absence of these chiral fermions. Here, the subscript indices label the
U(
1
)1, U(
1
)2 and U(
1
)′ charges.
To count the number of vector-like pairs for each representation, we have to calculate
the dimensions of its corresponding cohomology H (M, Cp) where Cp denotes the internal
bundle. For our purpose, it is sufficient to consider the number of chiral modes to derive
∗
the particles in the standard-like models. Recall that the Higgs and higgsino fields will be
identified with L43,,54 and/or their conjugates in the latter analysis. Although it is possible to
break the Pati-Salam-like symmetry by the vector-like Higgs (4¯, 1, 2, 1) + c.c., in this paper,
we concentrate on Wilson lines to break the Pati-Salam-like symmetry, namely SU(
4
)C →
SU(
3
)C ×U(
1
)3 and SU(
2
)R ×U(
1
)′ → U(
1
)4×U(
1
)5. It can be achieved by the introduction
of discrete Wilson lines on the quotient CICYs M˜ = M/Γ, where Γ should be different from
Z4 and Z2. However, in such a scenario, U(
1
)B−L gauge symmetry is still unbroken on the
quotient CICYs. One of the possibilities to break U(
1
)B−L gauge symmetry is the non-zero
vacuum expectation value of at least one of the right-handed sneutrino which carries the
U(
1
)B−L charge not the U(
1
)Y charge. When the soft mass of the right-handed sneutrino
is tachyonic at the TeV scale through renormalization group effects, the sneutrino develops
a vacuum expectation value. Such a spontaneous U(
1
)B−L breaking is demonstrated in
the B-L MSSM derived from the E8 × E8 heterotic string theory [41, 42]. This U(
1
)B−L
breaking scenario requires a deeper understanding of the mechanism of supersymmetry
breaking and the stabilization of moduli fields. It will be the subject of future work.
3.2
Explicit three-generation models
4,40
3 ×
,
Let us search for the models passing all the requirements on the quotient CICYs within
2 ≤ h1,1
are totally four quotient CICYs where one of the discrete symmetry is different from Z
≤ 5. For the 6 CICYs with h1,1(M) = 2 and 12 CICYs with h1,1(M) = 3, there
2
and Z4 [39]. They are characterized by the configuration matrices listed in appendix B.
However, one cannot find realistic models within the range |mia| ≤ 8.
In a similar way, we next search for the models on M with h1,1(M) = 4. Although
there exist two possible CICYs different from Z2 and Z4 for the 19 CICYs with h1,1(M) = 4,
the following CICY,
leads to realistic models. It admits the Z3 and Z
Z3 freely-acting discrete symmetries.
The nonvanishing intersection numbers and second Chern number of the tangent bundle
(3.6)
Q2 χ(M,L1−1 ⊗ L2)/|Γ|
HJEP05(218)4
Index
tries refer to the representations under (SU(
4
)C × SU(
2
)L × SU(
2
)R × SO(16))U(
1
)1,U(
1
)2,U(
1
)4,U(
1
)5
and (SU(
3
)C × SU(
2
)L × SO(16))U(
1
)1,U(
1
)2,U(
1
)3,U(
1
)4,U(
1
)5 and the subscript indices label the U(
1
)
charges. Note that in the first column, two U(
1
)4,5 charge vectors q = (q4,q5) and q′ = (q4′,q5′) are
Q1
L1
L2
c
uR1
c
dR1
¯c
dR2
n1
ecR15
expressed in terms of a basis w˜i on M are given by
d123 = 6,
respectively. Since one cannot find realistic models satisfying all the consistency conditions
within the range |mia| ≤ 4 for the Z3 ×
Z3 case, we focus on the Z3 case where the number
of complex structure moduli as well as the Euler number on the quotient CY manifold
M˜ = M/Z3 reduce to be h2,1 = 16 and χ = −24.
When we search for the line bundles within the range |mia| ≤ 4 to realize eqs. (3.4)
and (3.5), namely the three generations of quarks and leptons and no chiral exotics on
the quotient CICY M˜ , it turns out that the lists of the first Chern numbers in tables 4
and 5 yield realistic models. In both cases with tables 4 and 5, the rank of the mass
matrix of U(
1
)s in eq. (2.29) is 3 and the remaining two U(
1
) gauge symmetries are U(
1
)Y
and U(
1
)B−L = U(
1
)3/3. The distinction between the spectrum of tables 4 and 5 are the
existence of Higgs fields which could be identified with L43,,54 and/or their conjugates. The
indices of Higgs fields L43,,54 vanish for the choice of line bundles in table 4, whereas Higgs
fields appear in the case of table 5. As commented before, the exotic modes such as dR1 and
c
dR2
are the vector-like particles with respect to the standard model gauge group with an
extra U(
1
)B−L = U(
1
)3/3 on the quotient CICY. Indeed, the net-number of chiral modes is
the same because of χ(L22) = −χ(L2−2). We expect that such modes could become massive
due to the loop-effects for the anomalous U(
1
)2 symmetry. Since the obtained spectrum
illustrated in table 6 consists of the three generations of quarks and leptons, vector-like
Higgs and extra vector-like particles with respect to SU(3)C × SU(
2
)L × U(
1
)Y × U(
1
)B−L,
this models are free of gauge and gravitational anomalies. For other examples in table 5,
the particle spectrum is similar to that in table 6, but the flavor structure of Yukawa
couplings could be different from one another. Indeed, for some of the line bundles listed
in table 5, the number of chiral generation is determined by only Q1(L1) or Q2(L2) in
contrast to the case of table 6.
Let us take a closer look at the possible Yukawa couplings among the standard model
particles with an emphasis on the case of table 6. In this model, we find that the gauge
symmetries of the low-energy effective action allow for the following Yukawa couplings of
quarks and leptons,
(Q1, L¯35, ucR25),
(L1, L44, ecR14),
(Q2, L¯45, ucR25),
(L2, L34, ecR14),
(Q1, L44, dcR34),
(L1, L¯35, nc2 5),
(Q2, L34, dcR34),
(L2, L¯45, nc2 5),
(3.9)
by identifying the up-and down-type Higgs doublets as L¯53,4 and L43,4. Here, L¯54 stands for
the conjugate representation of L54 and those Higgs fields are vector-like particles under
the standard model gauge group with an extra U(
1
)B−L. Three right-handed neutrinos are
identified with n52. Although those couplings are allowed by the gauge symmetries, there
(∓2, 0, ±1, 0)
(
0, 1, −2, 0
)
(∓1, ±1, 0, 0)
(
6, 24, 12, 0
)
ci1(La) = mia in agreement with all the requirements, where other line bundles are constrained as
c1(L5) = −c1(L4) and c1(L3) = 0.
among the mia. The list of this table gives rise to the vanishing indices of Higgs fields L43,,54.
We omit the other sign flipping and possible permutations
(m11, m21, m31, m41) (m12, m22, m32, m42) (m14, m24, m34, m44)
# of Five-branes
(∓3, 0, 0, ±3)
(∓1, 0, ±1, ∓3)
(∓1, 0, ±1, ∓3)
(∓1, 0, ±1, ∓2)
(
1, 0, −1, 2
)
(
0, 1, −1, −2
)
(
1, −1, 0, −1
)
(
0, 1, −1, −1
)
(0, ∓1, ±1, ±4)
(∓1, ±1, 0, ±1)
(0, ∓1, ±1, ∓1)
(∓1, ±1, 0, 0)
(
30, 48, 30, 0
)
(42, 75, 51, 12)
(39, 27, 66, 12)
(
18, 45, 33, 12
)
among the mia. One of the chiral spectrum is illustrated in table 6.
ci1(La) = mia in agreement with all the requirements, where other line bundles are constrained as
c1(L5) = −c1(L4) and c1(L3) = 0. We omit the other sign flipping and possible permutations
exist topological selection rules for their Yukawa couplings as discussed in refs. [43–45].
We leave the detailed study of the Yukawa couplings for a future work. With the help
of U(
1
)B−L and other anomalous U(
1
) symmetries, R-parity violating and the
dimensionfour and -five proton decay operators such as QQQL and ucucdcec are forbidden. As stated
already, we will also postpone the detailed study of U(
1
)B−L breaking for a future analysis.
For the 23 CICYs with h1,1 = 5, we have searched for the three-generation models and
among them, there are four possible candidates where one of the discrete symmetry group
is different from Z
2 and Z4 [39]. However, it turns out that the line bundle background
does not pass all the requirements within the range |mia| ≤ 4.
4
Conclusion
We have searched for the three-generation standard-like models from the SO(32) heterotic
string theory on quotient complete intersection Calabi-Yau manifolds with line bundles.
The line bundles with U(
1
) structure groups, in contrast to non-abelian bundles, are
promising tool to realize standard-like models not only in the E8 × E8 heterotic string,
but also the SO(32) one. Such stable line bundles lead to the three chiral generations of
quarks and leptons.
In this paper, we have presented that the Wilson lines and line bundles satisfying
the supersymmetric conditions, the U(
1
)Y masslessness conditions and K-theory condition
yield the standard-like model gauge group SU(3)C × SU(
2
)L × U(
1
)Y × U(
1
)B−L via a
Pati-Salam-like symmetry. Other U(
1
) gauge bosons become massive through the
GreenSchwarz mechanism. The obtained spectrum contains the three generations of quarks and
leptons, vector-like Higgs and extra vector-like particles with respect to SU(
3
)C × SU(
2
)L ×
HJEP05(218)4
-1
-2
-1
-2
-4
-28
28
4
0
0
-3
-12
12
-3
0
-3
-3
-24
0
0
0
0
0
0
0
0
0
0
-3
-3
-3
-3
-3
0
0
-3
-24
0
0
0
0
0
0
HJEP05(218)4
Q1
Q2
L1
L2
L4
3
L5
3
L4
4
L5
4
c
uR1
ucR24
ucR25
c
dR1
c
dR2
dcR34
dcR35
ecR14
ecR15
Supersymmetric spectrum in visible and hidden sectors for the line bundles
(m11,m21,m31,m41)
=
(
−3,0,0,3
), (m12,m22,m32,m42)
=
(
1,0,−1,2
) and (m14,m24,m34,m44) =
(
0,−1,1,4
) in table 5. In the second column, the entries refer to the representations under SU(3)C,
SU(2)L and SO(16) and the subscript indices label the U(1) charges.
U(
1
)Y × U(
1
)B−L. Although the Yukawa couplings are allowed by the gauge symmetries of
the low-energy effective action, we leave the derivation of selection rules for the holomorphic
Yukawa couplings on CICYs [44–46] for a future work.
Motivated by S- and T-dual models of intersecting D-branes in type IIA string theory,
we have focused on the Pati-Salam-like gauge group to derive the standard model one. It
would be interesting to check whether SO(32) gauge group is directly broken to the standard
model gauge group by a hypercharge flux, although there exists a no-go theorem on the
existence of a standard model spectrum consistent with the hypercharge flux breaking and
gauge coupling unification in the E8 × E8 heterotic string [47]. The gauge couplings of
nonabelian gauge groups embedded in the SO(32) heterotic string are non-universal in contrast
to those in the E8 × E8 heterotic string [10, 11]. Such non-universal gauge couplings have
the potential to explain the differences between the gauge couplings of SU(
3
)C and SU(
2
)L
at the string scale as demonstrated in the toroidal background [48].
Acknowledgments
We would like to thank H. Abe, T. Kobayashi, Y. Takano, T. H. Tatsuishi and A. Lukas
for useful discussions and comments. We also would like to thank the referee for valuable
comments to improve the paper and organizing committee of the conference of “String
Phenomenology 2017” for their hospitality. H. O. was supported in part by Grant-in-Aid
for Young Scientists (B) (No. 17K14303) from Japan Society for the Promotion of Science.
A
Net-number of chiral fields
In this appendix, we present the index formula to calculate the net-number of chiral fields
in table 3,
χ(Lpa) =
Z
M
ch3(Lpa) + c2(T M)c1(Lpa)
12
= dijk 6
p
3
miamjamka +
p
12 miacj2k(T M) ,
χ(Lpa ⊗ Lbq) = χ(Lpa) + χ(Lbq) +
c1(Lpa)ch2(Lbq) + ch2(Lpa)c1(Lbq)
Z
1
M
= χ(Lpa) + χ(Lbq) +
2 dijk pq2miamjbmbk + p2qmiamjambk ,
(A.1)
where ci1(La) = mia and p, q, r are the integers.
B
List of CICYs to be applicable for the Pati-Salam models
the second Chern number in the basis wˆi.
To break the Pati-Salam-like symmetry by the inclusion of Wilson lines, the freely-acting
discrete symmetry groups of CICYs should be different from Z
4 and Z2. We list such
topologically distinguishable CICYs for 2 ≤ h1,1 ≤ 5 in table 7 [39], where c2(T M) denotes
HJEP05(218)4
(h1,1, h2,1)
Configuration matrix
P2 "0 1 1 1#
P10 0 0 2
P10 2 0 0
P10 0 2 0
P12 0 0 0
aAlthough this tetra-quadric has other discrete symmetries Z4, Z2 × Z2, Z4 × Z2, Q8, Z4 × Z4, Z4 ⋊ Z4,
Z8 × Z2, Z8 ⋊ Z2, Z2 × Q8, we focus on the Z8 case.
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
HJEP05(218)4
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