Non-Abelian discrete flavor symmetries of 10D SYM theory with magnetized extra dimensions

41 Pages·2014·1.82 MB·English

Published for SISSA by Springer
Received: April 7, 2014
Accepted: May 2, 2014
Published: June 3, 2014
Non-Abelian discrete flavor symmetries of 10D SYM
theory with magnetized extra dimensions J
H
E
P
Hiroyuki Abe,a Tatsuo Kobayashi,b,c Hiroshi Ohki,d Keigo Sumitaa and 0
Yoshiyuki Tatsutaa
6
aDepartment of Physics, Waseda University,
(
Tokyo 169-8555, Japan
2
bDepartment of Physics, Kyoto University,
Kyoto 606-8502, Japan 0
cDepartment of Physics, Hokkaido University, 1
Sapporo 060-0810, Japan
4
dKobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI),
Nagoya University, )
Nagoya 464-8602, Japan 0
E-mail: [email protected], [email protected],
1
[email protected], [email protected],
7
y [email protected]
Abstract: We study discrete flavor symmetries of the models based on a ten-dimensional
supersymmetric Yang-Mills (10D SYM) theory compactified on magnetized tori. We as-
sumenon-vanishingnon-factorizablefluxesaswellastheorbifoldprojections. Thesesetups
allow model-building with more various flavor structures. Indeed, we show that there exist
various classes of non-Abelian discrete flavor symmetries. In particular, we find that S
3
flavor symmetries can be realized in the framework of the magnetized 10D SYM theory for
the first time.
Keywords: Field Theories in Higher Dimensions, Discrete and Finite Symmetries
ArXiv ePrint: 1404.0137
Open Access, (cid:13)c The Authors.
doi:10.1007/JHEP06(2014)017
Article funded by SCOAP3.
Contents
1 Introduction 1
2 Magnetized brane models 3
2.1 Magnetized torus model with factorizable fluxes 4
2.2 Magnetized torus model with non-factorizable fluxes 6
2.3 Magnetized orbifold model with non-factorizable fluxes 9
J
3 Degenerated structures of zero-modes 10
H
3.1 Generation-types 10
E
3.2 The relation between generation-types in each sectors 13
P
3.3 Exceptional generation-types in magnetized orbifold model 13
0
4 Non-Abelian discrete flavor symmetry on magnetized brane models 14
6
4.1 Magnetized torus model with factorizable fluxes 14
(
4.2 Magnetizedtorusmodelwithnon-factorizablefluxes: alignedgeneration-types 15
2
4.3 Magnetized torus model with non-factorizable fluxes: not-aligned
generation-types 23 0
4.4 Magnetized orbifold model with non-factorizable fluxes 25 1
4
5 Non-Abelian discrete flavor symmetry from gauge symmetry breaking 26
)
6 Conclusions and discussions 30
0
A The generation-types for detN = n 32 1
7
B More about flavor symmetries in three-generation models: aligned gen-
eration-types 34
C Flavor symmetries in four-generation models 34
1 Introduction
The standard model (SM) of particle physics is a quite successful theory, which can explain
experimental data so far. However, there are still several mysteries and puzzles. For
example, the SM has many free parameters including the neutrino masses. Most of such
free parameters appear in Yukawa couplings of quarks and leptons, i.e., in the flavor sector.
Recent experiments of neutrino oscillations reported relatively large mixing angles in the
lepton sector. They are completely different from the quark mixing angles. Therefore, it is
quiteimportanttostudyarealisticandnaturalmodelthatcansimultaneouslyexplainsuch
– 1 –
mixingpatternsofquarksandleptons. AcertainsymmetrycouldcontrolYukawacouplings
among three generations. Indeed, quark and lepton masses and mixing angles have been
studied from the viewpoint of flavor symmetries, in particular non-Abelian discrete flavor
symmetries [1–5].
Superstringtheoryisapromisingcandidatefortheunifiedtheoryofalltheinteractions
includinggravity and all the matter fields and Higgs fields. Superstring theory is defined in
ten-dimensional (10D) spacetime and then predicts extra six dimensions compactified on
somecompactspaceinadditiontotheobservedfour-dimensional(4D)spacetime. Further-
more, supersymmetric Yang-Mills (SYM) theory in higher dimensional spacetime appears
as effective field theory of superstring theory. That leads to quite interesting aspects from J
both theoretical and phenomenological points of view. (See ref. [6] for a review of super- H
string phenomenology.) It is important to study the structure of such an internal compact
E
space, especially, fromthelatterviewpoint. Thedetailedstructureoftheinternalspacede-
P
terminesimportantaspectsofparticlephenomenologyinfour-dimensional(4D)low-energy
0
effective field theory (LEEFT), e.g., mass spectra including the generation number, cou-
pling selection rules, coupling strength, symmetries in 4D LEEFT, etc. For example, the 6
toroidal compactification is one of the simplest compactifications, but 10D SYM theory on (
the 6D torus without any gauge background as well as superstring theory leads to = 4
N 2
supersymmetry in 4D spacetime. That is non-chiral theory and not realistic. The orbifold
0
compactificationandthetoruscompactificationwithmagneticfluxesaswellastheorbifold
1
with magnetic fluxes can reduce the number of 4D supersymmetric currents and lead to
4D chiral theory. Thus, these are quite interesting to study. 4
Recently, magnetic fluxes in extra dimensions have been receiving many attentions.1 )
The = 4 supersymmetry is broken by magnetic fluxes down to = 0, 1 or 2, which
0
N N
depends on the configuration of magnetic fluxes. It is quite interesting that the simplest
1
toroidal compactifications with magnetic fluxes in extra dimensions lead to 4D chiral spec-
7
tra, starting from higher-dimensional SYM theories which might be obtained as LEEFT of
superstringtheories[8–12]. Inaddition,thestructureofcompactsixdimensionsdetermines
generations of chiral matters, masses and couplings of the 4D LEEFT after dimensional re-
ductions. For example, the degeneracy of chiral zero-modes, i.e., the number of generation,
is determined by the magnitude of magnetic fluxes, and the overlap integrals of localized
zero-mode wavefunctions yieldYukawa couplings for chiral matter fields in the 4D LEEFT.
Indeed, manyphenomenologicallyimportantpropertiesoftheSM,suchasthe4Dchirality,
the number of generations and hierarchical Yukawa couplings [13, 14] could be originated
from the magnetic fluxes.
Furthermore, it is known that magnetized D-brane models as well as intersecting
D-brane models can derive certain non-Abelian discrete flavor symmetries such as D ,
4
∆(27) and ∆(54) [13, 15–22].2 Similar flavor symmetries can be obtained from heterotic
orbifold models [24–26]. Thus, non-Abelian discrete symmetries which play a role in
particle physics can arise from the underlying theory, e.g., superstring theory. In addition,
1See for a review of phenomenological aspects in orbifold compactification [7].
2See also ref. [23].
– 2 –
non-Abelian discrete symmetries are interesting ideas for controlling flavor structures
in model-building in the bottom-up approach as mentioned above. These could provide
a bridge between the low-energy phenomenology and the underlying theory, especially
superstring theory. Therefore, it is interesting and important to study the non-Abelian
discrete flavor symmetry obtained from the magnetized brane models as the low-energy
effective theory of superstring theory.
Inourpreviouspaper[27], westudiedtheflavorstructuresrealizedbynon-factorizable
fluxesontoroidalextradimensions. Thatexpandedthepossibilitiesfornewtypesofmodel
building, and indeed we have obtained several new types of models with the SM particle
content as massless modes. Then, it turned out that non-factorizable fluxes can lead rich J
flavor structures in three-generation models of quarks and leptons. Because of these facts, H
it is quite attractive to study the flavor symmetry realized in the magnetized models with
E
the extension to non-factorizable fluxes.
P
This paper is organized as follows. In section 2, we review the magnetized 10D SYM
0
theory and the fields appearing in its action. In addition, we explain the chiral zero-
modes and Yukawa couplings in two cases with factorizable and non-factorizable fluxes, 6
respectively. Thenwedevelopawaytolabelthezero-modeswithnon-factorizablefluxesin (
detailinsection3. Insection4,weshowthenon-Abeliandiscreteflavorsymmetriesrealized
2
in the 10D SYM theory with generic configurations of magnetic fluxes in extra dimensions.
0
Inaddition,weconfirmthattheseflavorsymmetriescouldberederivedfromtheperspective
1
ofthenon-Abeliandiscretegaugesymmetry,insection5. Section6isdevotedtodiscussions
andconclusions. InappendixA,werefertothenumberofgeneration-typesforthearbitrary 4
degeneracy of zero-modes and give some interpretations for them. In appendix B and C,
)
we enumerate and discuss the examples of some configurations of magnetic fluxes in three-
0
and four-generation models, respectively, which are not explained in section 4.
1
7
2 Magnetized brane models
Westartwith10DSYMtheory. Weconsider4DflatMinkowskispacetimeandfactorizable
three tori T2 T2 T2, that is, R3,1 (T2)3. The Lagrangian is given by
× × ×
1 i
= Tr FMNF + Tr λ¯ΓMD λ , (2.1)
L −4g2 MN 2g2 M
(cid:0) (cid:1) (cid:0) (cid:1)
where g is a 10D YM gauge coupling constant and M,N = 0,1,...,9. The field strength
F and the covariant derivative D are written by
MN M
F = ∂ A ∂ A i[A ,A ], (2.2)
MN M N N M M N
− −
D λ = ∂ λ i[A ,λ]. (2.3)
M M M
−
In the following, we use x and y as two real coordinates on the i-th T2 for i = 1,2,3. The
i i
SYM theory includes a 10D vector field A and a 10D Majorana-Weyl spinor field λ. The
M
trace in the above Lagrangian acts the indices of YM gauge group.
– 3 –
For convenience we adopt complex coordinates z and complex vector fields A for
i i
i = 1,2,3, which are defined as
1 1
z = (x +τ y ), A = (A τ¯A ). (2.4)
i i i i i 3+2i i 2+2i
2 Imτ −
i
The10DSYMtheorypossesses = 4supersymmetrycountedintermsof4Dsupercharges.
N
The 10D vector field A and Majorana-Weyl spinor field λ are decomposed into 4D = 1
M
N
single vector and triple chiral multiplets, i.e., V = A ,λ and φ = A ,λ (i = 1,2,3).
µ 0 i i i
{ } { }
For 4D positive chirality, these spinor fields λ , λ , λ and λ have the 6D chiralities,
0 1 2 3
(+,+,+), (+, , ), ( ,+, ), and ( , ,+) on 6D spacetime R3,1 T2 for i = 1,2,3,
− − − − − − × i
respectively. The 4D = 1 single vector and triple chiral multiplets can be expressed in J
N
terms of vector superfield V and chiral superfields φi (i = 1,2,3). H
E
2.1 Magnetized torus model with factorizable fluxes
P
We consider the 10D SYM theory with two types of magnetic fluxes, factorizable flux and
0
non-factorizable flux. In this subsection, we review the former factorizable case based on
ref. [12], and assume the following magnetic background: 6
A = π M(i)z¯ +ζ¯ , A = λ = λ = 0, (2.5) (
i i i µ 0 i
h i Imτ h i h i h i
i 2
(cid:16) (cid:17)
where M(i) and ζ are N N matrices of (Abelian) magnetic fluxes and Wilson-lines,
i × 0
respectively,3 given as
1
(i) (i)
M 1 ζ 1
1 N1 1 N1 4
(i) (i)
 M 1   ζ 1 
M(i) = 2 N2 , ζ = 2 N2 , (2.6) )
 ...  i  ...  0
   
 (i)   (i) 
 Mn 1Nn  ζn 1Nn 1
   
with a positive integer N (a = 1,2,...,n) satisfying n N = N, and τ denotes the 7
a a=1 a i
complex structure parameter that characterizes the shape of the i-th T2. When there are
P
non-vanishingmagneticfluxesandWilson-lines,theformofVEV(2.5)leadstofactorizable
fluxes. Here, the magnetic fluxes satisfy M(i),M(i),...,M(i) Z due to Dirac’s quanti-
1 2 n ∈
(i) (i) (i)
zation condition. In the case that the magnetic fluxes M ,M ,...,M take different
1 2 n
values from each other, U(N) gauge group breaks into U(N ) U(N ) ... U(N ). The
1 2 n
× × ×
same holds for Wilson-lines. We use indices a,b,... for labeling the unbroken subgroups
U(N ),U(N ),... of U(N), respectively. The block off-diagonal part (φ ) of chiral super-
a b i ab
field φ is the bi-fundamental representation under U(N ) U(N ) and the block diagonal
i a b
×
part (φ ) is the adjoint representation under U(N ).
i aa a
Next, we refer to the zero-mode equations for chiral matter superfields φ . The zero-
j
mode equations are found as [28]
1
∂¯f(i)+ A¯ ,f(i) = 0 (i = j), (2.7)
i j √2 h ii j
h i
3Forsimplicity,weassumethefollowingformsofmagneticfluxesandWilson-lines,althoughthosearenot
general forms. In general, we can choose the different forms of magnetic fluxes and Wilson-lines from each
torusT2. However,wedonotrequiresuchgeneralformswhenwestudyonlynon-Abelianflavorsymmetries.
i
– 4 –
1
(i) (i)
∂ f A ,f = 0 (i = j), (2.8)
i j − √2 h ii j 6
h i
where f(i) denotes the zero-mode wavefunction of the chiral superfield φ on the i-th T2
j j
and then ∂ ∂/∂z . Note that a difference of the signs in eqs. (2.7) and (2.8) comes
i i
≡
(i)
from the chirality structure. The zero-mode wavefunctions f , in general, satisfy different
j
equations on each T2 for i = 1,2,3. Due to the existence of non-vanishing magnetic fluxes,
i
chirality projection occurs.
(i=j) (i) (i) (i)
For the zero-mode wavefunction f in the ab-sector, if M M M > 0,
j ab ≡ a − b
(i)
then there exist M solutions of zero-mode equation (2.7),
| ab | J
H
(f(i)) = gΘIa(ib),Ma(ib)(z′) (I(i) = 1,2,..., M(i) ), (2.9)
j ab j i ab | ab | E
(i) (i)
I /M
ΘjIa(ib),Ma(ib)(zi) = NeiπMa(ib)ziImzi/Imτi ·ϑ ab ab (Ma(ib)zi,Ma(ib)τi), (2.10) P
0 0
  6
where z′ z +ζ(i)/M(i), ζ(i) ζ(i) ζ(i), and ϑ denotes Jacobi ϑ-function,
i ≡ i ab ab ab ≡ a − b (
2
a
ϑ (ν,τ) = eπi(a+l)2τe2πi(a+l)(ν+b). (2.11) 0
 
b l∈Z
X 1
 
(i) 4
On the other hand, there is no normalizable zero-mode wavefunction if M < 0 and,
ab
(i) )
finally, the zero-mode wavefunction is constant if M = 0.
ab
For the zero-mode wavefunction f(i6=j) in the ab-sector, if M(i) < 0, the zero-mode 0
j ab
wavefunctions can be written as the complex conjugate of the wavefunction (2.9). There 1
(i)
is no normalizable zero-mode wavefunction if M > 0 and the zero-mode wavefunction is
ab 7
(i)
constant if M = 0. Notice that the degeneracy of the zero-modes in the chiral superfield
ab
(i)
φ on the i-th torus is determined by the number of the magnetic fluxes M that the φ
j ab j
feels on the i-th torus. With three toroidal compactifications, the total degeneracy N of
ab
the chiral zero-modes in (φ ) can be written by N = 3 M(i) .
j ab ab i=1 ab
The Yukawa couplings between chiral zero-modes in the 4D effective theory are given
Q (cid:12) (cid:12)
(cid:12) (cid:12)
by the overlap integrals,
3
λ = d2z detg(i)(f(i)) (f(i)) (f(i)) , (2.12)
IJK i 1 ab 3 bc 2 ca
i=1Z q
Y
where g(i) denotes the metric for the i-th torus T2 and (I(1),I(2),I(3)) labels the
i I ≡ ab ab ab
total generation of zero-modes in ab-sector. The same holds for the other sectors. We can
(i) (i) (i)
calculate the overlap integral (2.12) under M +M +M = 0, which are evaluated as
ab bc ca
3
λ = λ , (2.13)
IJK Ia(ib)Ic(ai)Ib(ci)
i=1
Y
– 5 –
Mc(ai)
λ δ (2.14)
Ia(ib)Ic(ai)Ib(ci) ∝ Ia(ib)+Ic(ai)+Ma(ib)m,Ib(ci)
m=1
X
Mc(ai)Ia(ib)−Ma(ib)Ic(ai)+Ma(ib)Mc(ai)m
×ϑ −Ma(ib)Mc(ai)Mb(ci) (Mb(ci)ζ¯c(ai)−Ma(ib)ζ¯b(ci),−τ¯iMa(ib)Mc(ai)Mb(ci)),
0
 
 
where we omit an overall factor, because the factor has no effect on the flavor symmetry
in magnetized torus models.
J
2.2 Magnetized torus model with non-factorizable fluxes
H
Next, we review the generalization of the above results including non-factorizable fluxes,
based on refs. [12, 29, 30]. We assume the following magnetic background, E
P
π
A = M(i)z¯ +M(ij)z¯ +ζ¯ , (2.15)
h ii Imτ i j i 0
i
(cid:16) (cid:17)
Aµ = λ0 = λi = 0, (2.16) 6
h i h i h i
(
with i = j, where M(ij) is a N N matrix of an additional (Abelian) magnetic fluxes,
6 ×
2
(ij)
M 1
1 N1 0
(ij)
M(ij) =  M2 1N2 , (2.17) 1
 ...  4
 
 (ij) 
 Mn 1Nn′ )
 
0
with a positive integer Na′ (a′ = 1,2,...,n′) satisfying na′′Na′ = N.4 It holds that
1
M(ij),M(ij),...,M(ij) Z due to Dirac’s quantization condition. The magnetic back-
1 2 n ∈ P
7
ground (2.15) is a straightforward extension of eq. (2.5) and leads to non-factorizable
magnetic fluxes.
We substitute the VEVs (2.15) into zero-mode equations (2.7) and (2.8) and find
that the zero-mode wavefunctions and the degeneracy of zero-modes are changed from the
factorizable case. Again, we focus on chiral superfields φ (i = 1,2,3) and then explain
i
their zero-mode wavefunctions in the following. In this paper, we consider the case that
only magnetic fluxes M(12) and M(21) in the first and the second tori T2 T2 are turned
1 × 2
on. The extensions to the other non-vanishing magnetic fluxes M(ij) are straightforward.
Now, we define the matrix
(1) (21)
M M
N ab ab , (2.18)
ab ≡ M(12) M(2)!
ab ab
M(i) M(i) M(i), (2.19)
ab ≡ a − b
Imτ
M(ij) i(M(ij) M(ij))+(M(ji) M(ji)), (2.20)
ab ≡ Imτ a − b a − b
j
4As mentioned in eq. (2.6), these magnetic fluxes and Wilson-lines are not general forms.
– 6 –
whichdeterminesthedegeneracyofzero-modes. Notethatdiagonalelementsofthematrix
N correspond to the magnetic fluxes defined in eq. (2.6).
ab
Next,inordertoobtainthenormalizablewavefunctionswiththematrixNandcomplex
structure parameters τ (i = 1,2), we must impose the Riemann conditions
i
Nij Z, (N ImΩ)T = N ImΩ, N ImΩ > 0, a,b, (2.21)
ab ∈ ab· ab· ab· ∀
where Ω diag(τ ,τ ) is a 2 2 matrix constructed from complex structure parameters.
1 2
≡ ×
For a while, we consider the case with vanishing Wilson-lines, i.e., ζ¯ = ζ¯ = 0. Only if
1 2
the matrix N and the complex structure Ω satisfy the Riemann conditions (2.21), there
ab J
exist the normalizable zero-mode wavefunctions in the ab-sector on the first and second
H
tori, which are expressed as
E
(fj(12))ab = gΘ~jiab,Nab(~z), (2.22) P
~i 0
Θ~iab,Nab(~z) = eπi(Nab·~z)·(ImΩ)−1·(Im~z) ϑ ab (N ~z,N Ω), (2.23)
j N ·   ab· ab· 6
0
  (
where ~z (z ,z ) and ϑ denotes the Riemann ϑ-function,
1 2 2
≡
0
~a
ϑ (~ν,Ω) = eπi(~l+~a)·Ω·(~l+~a)e2πi(~l+~a)·(~ν+~b). (2.24) 1
~b
~lX∈Z2 4
 
Thevector~i labelsdegeneratedzero-modes(generations),andwewillexplainitsmeaning )
ab
in detail in the next section. 0
Note that the expression of the wavefunction (2.22) is for (totally) positive chirality 1
matters, which namely have the chirality (+,+) and ( , ) on the first and second tori.
− − 7
In 10D SYM theory with the superfield description [28] we adopt in this paper, the
wavefunction (2.22) is valid for a chiral superfield φ that has the chirality ( , ) on the
3
− −
first and second tori. For chiral superfields φ and φ , they need to be mixed up to be the
1 2
solution of the zero-mode equations. As stated in refs. [29, 30], we consider the following
parameterizations,
(φ ) = α Φ , (φ ) = β Φ . (2.25)
1 ab ab ab 2 ab ab ab
The Riemann conditions (2.21) to obtain normalizable zero-mode wavefunctions and an
explicit form of the zero-mode wavefunction (2.22) can be also applied for Φ by replacing
ab
the complex structure Ω with the effective complex structure Ω˜ Ωˆ Ω, where
ab
≡ ·
1 1 q2 2q
Ωˆ − ab − ab . (2.26)
ab ≡ 1+qa2b −2qab qa2b−1!
Mixing parameters q β /α are given for individual bi-fundamental representations
ab ab ab
≡
labeled by a and b (a = b), and their values are determined by the second condition of the
6
Riemann conditions.
– 7 –
On the third torus, the zero-mode wavefunction is the same as the expression (2.9)
or the complex conjugate to that. Thus, the degeneracy of zero-modes N with non-
ab
factorizablefluxesisdeterminedbythematrixN andthefluxM(3), i.e., N = detN
ab ab ab | ab×
(3) (3)
M for M = 0 in the present situation.
ab | ab 6
Next, the Yukawa couplings in the 4D effective theory is also evaluated by the overlap
integral
λ = λ d2z d2z detg(1)g(2)(f(12)) (f(12)) (f(12)) , (2.27)
IJK Ia(3b)Ic(a3)Ib(c3) 1 2 1 ab 3 bc 2 ca
Z q
where (~i , I(3))labelsthetotalgenerationofzero-modesinab-sector. Thesameholds J
I ≡ ab ab
fortheothersectors. Weconsiderthecasethattherearezero-modeswiththetotalnegative H
chirality on the first and second tori. Then, we can calculate the overlap integrals (2.27)
E
under N +N +N = 0 and M(3)+M(3)+M(3) = 0 [12, 29, 30], which are evaluated as
ab bc ca ab bc ca P
λ = λ λ , (2.28) 0
IJK ~iab~ica~ibc · Ia(3b)Ic(a3)Ib(c3)
6
λ~iab~ica~ibc ∝ δ~ibc,N−cb1(Nab~iab+Nca~ica+Nabm~) (
m~
X
K~ 2
dy dy e−π~y·(NabΩ˜ab+NcaΩ˜ca+NbcΩ)·~y ϑ (iY~ iQ~) , (2.29)
× 1 2 ·   |  0
Z 0
1
   
where ~y (y ,y ) and m~ denote the integer points in the region spanned by
1 2 4
≡
e~′ ~e (detN detN )N−1(N +N )N−1, (2.30) )
i ≡ i ab ca ca ab ca ab
0
1 0
~e1 = , ~e2 = , (2.31) 1
0! 1!
7
and
~i
K~ bc , (2.32)
≡ (~iab−~ica+m~)Nabd(eNtaNba+bNdceat)N−c1aNca !
(N Ω˜ +N Ω˜ +N Ω) ~y
Y~ ab ab ca ca bc · , (2.33)
≡ (detNabdetNca)(NabΩ˜ab(N−ab1)T −NcaΩ˜ca(N−ca1)T)·~y!
N Ω˜ +N Ω˜ +N Ω (detN detN )(N Ω˜ (N−1)T N Ω˜ (N−1)T)
Q~ ab ab ca ca bc ab ca ab ab ab − ca ca ca .
≡ (detNabdetNca)(Ω˜ab−Ω˜ca) (detNabdetNca)2(Ω˜abN−ab1+Ω˜caN−ca1) !
(2.34)
In eq. (2.29), again, we omit an overall factor, because of the same reason as the model
with factorizable fluxes in the previous subsection. Note that the integrals over z and z
1 2
are non-factorizable, while the one over z is factorized in the Yukawa couplings (2.27), as
3
a consequence of the flux configuration assumed above. The overlap integral on the third
torus yields the factor λ that is exactly the same as eq. (2.14) for i = 3. The
Ia(3b)Ic(a3)Ib(c3)
– 8 –
property of non-factorizable fluxes appears in the overlap integral on the first and second
tori. Therefore it is interesting to investigate the factor λ in eq. (2.29).
~iab~ica~ibc
We have limited the above discussion to the case with vanishing Wilson-lines. In
this paragraph, we show the zero-mode wavefunction and the Yukawa coupling with non-
vanishing Wilson-lines, i.e., ζ¯ ,ζ¯ = 0. Indeed, by means of shifting the coordinates, such
1 2
6
a zero-mode wavefunction can be obtained as
(f(12)) = gΘ~iab,Nab(z~′), (2.35)
j ab j
where z~′ ~z +N−1 ζ~ and ζ~ (ζ(1),ζ(2)). By calculating the overlap integral of the
≡ ab · ab ab ≡ ab ab J
abovezero-modewavefunctionsonthefirstandsecondtori,therelevantpartoftheYukawa
H
couplings in the 4D effective theory can be obtained as
E
λ~iab~ica~ibc ∝ δ~ibc,N−bc1(Nab~iab+Nca~ica+Nabm~) (2.36) P
m~
X 0
K~
dy1dy2 e−π(y~′ab·NabΩ˜ab·y~′ab+y~′ca·NcaΩ˜ca·y~′ca+y~′bc·NbcΩ·y~′bc) ϑ (iY~ iQ~) , 6
×  ·   | 
Z 0 (
   
2
up to an overall factor. Moreover, we should replace Y~ in eq. (2.36) with
0
N Ω˜ y~′ +N Ω˜ y~′ +N Ω y~′ 1
Y~ ab ab· ab ca ca· ca bc · bc , (2.37)
≡ (detNabdetNca)(NabΩ˜ab(N−ab1)T ·y~′ab−NcaΩ˜ca(N−ca1)T ·y~′ca)! 4
)
wherewedefiney~′ ~y +(N ImΩ˜ )−1 Imζ~ forthezero-modewavefunction(f(12)) .
ab ≡ ab ab ab · ab 1 ab 0
(12)
The same holds for the zero-mode wavefunction (f ) . For the ca-sector in chiral su-
2 ca
perfield φ , we replace ~y with y~′ ~y +(N ImΩ)−1 Imζ~ . 1
3 bc bc ≡ bc bc · bc
7
2.3 Magnetized orbifold model with non-factorizable fluxes
Finally in this section we review the orbifold projection with non-factorizable fluxes based
on ref. [27]. In our previous paper [27], we extend the model proposed in ref. [31] (see also
ref.[17])wheretheorbifoldmodelswithfactorizablefluxesareconstructed. Thenumberof
the (degenerate) zero-modes is changed by the orbifold projection. We consider the T6/Z
2
orbifold where the Z projection acts on the first and second tori. It is constructed by
2
dividing T6 by the Z projection z z and z z , simultaneously. Such an iden-
2 1 1 2 2
→ − → −
tification prohibits (continuous) non-vanishing Wilson-lines. Here, we consider vanishing
Wilson-lines. On such an orbifold, we impose the following boundary conditions for 10D
superfields V and φ ,
i
V(x , z , z ,z ) = +PV(x ,z ,z ,z )P−1, (2.38)
µ 1 2 3 µ 1 2 3
− −
φ (x , z , z ,z ) = Pφ (x ,z ,z ,z )P−1, (2.39)
1 µ 1 2 3 1 µ 1 2 3
− − −
φ (x , z , z ,z ) = Pφ (x ,z ,z ,z )P−1, (2.40)
2 µ 1 2 3 2 µ 1 2 3
− − −
φ (x , z , z ,z ) = +Pφ (x ,z ,z ,z )P−1, (2.41)
3 µ 1 2 3 3 µ 1 2 3
− −
– 9 –