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Overview of Extra Dimensional Models with Magnetic Fluxes Tatsuo Kobayashi. 1. Introduction 2. Magnetized extra dimensions 3. N-point couplings and flavor symmetries 4 . Summary based on collaborations with H.Abe, K.S.Choi, R.Maruyama, M.Murata, Y.Nakai, H.Ohki, M.Sakai, K.Sumita.

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overview of extra dimensional models with magnetic fluxes tatsuo kobayashi
Overview of Extra Dimensional Models with Magnetic Fluxes Tatsuo Kobayashi


2. Magnetized extra dimensions

3. N-point couplings and flavor symmetries


based on collaborations with

H.Abe, K.S.Choi, R.Maruyama, M.Murata, Y.Nakai,

H.Ohki, M.Sakai, K.Sumita

1 Introduction

Extra dimensional field theories,

in particular

string-derived extra dimensional field theories,

play important roles in particle physics

as well as cosmology .

extra dimensions
Extra dimensions

4 + n dimensions

4D ⇒our 4D space-time

nD  ⇒ compact space

Examples of compact space

torus, orbifold, CY, etc.

field theory in higher dimensions
Field theory in higher dimensions

10D  ⇒ 4D our space-time + 6D space

10D vector

4D vector + 4D scalars

SO(10) spinor ⇒SO(4) spinor

x SO(6) spinor

internal quantum number

several fields in higher dimensions
Several Fields in higher dimensions

4D (Dirac) spinor

⇒(4D) Clifford algebra

(4x4) gamma matrices

represention space ⇒ spinor representation

6D Clifford algebra

6D spinor

6D spinor   ⇒  4D spinor x (internal spinor)

internal quantum


field theory in higher dimensions1
Field theory in higher dimensions

Mode expansions

KK decomposition

kk docomposition on torus
KK docomposition on torus

torus with vanishing gauge background

Boundary conditions

We concentrate on zero-modes.

zero modes

Zero-mode equation

⇒non-trival zero-mode profile

the number of zero-modes

4d effective theory
4D effective theory

Higher dimensional Lagrangian (e.g. 10D)

integrate the compact space ⇒4D theory

Coupling is obtained by the overlap

integral of wavefunctions

couplings in 4d
Couplings in 4D

Zero-mode profiles are quasi-localized

far away from each other in compact space

⇒suppressed couplings

chiral theory
Chiral theory

When we start with extra dimensional field theories,

how to realize chiral theories is one of important issues from the viewpoint of particle physics.

Zero-modes between chiral and anti-chiral

fields are different from each other

on certain backgrounds,

e.g. CY, toroidal orbifold, warped orbifold,

magnetized extra dimension, etc.

magnetic flux
Magnetic flux

The limited number of solutions with

non-trivial backgrounds are known.

Generic CY is difficult.

Toroidal/Wapred orbifolds are well-known.

Background with magnetic flux is

one of interesting backgrounds.

magnetic flux1
Magnetic flux

Indeed, several studies have been done

in both extra dimensional field theories

and string theories with magnetic flux


In particular, magnetized D-brane models

are T-duals of intersecting D-brane models.

Several interesting models have been

constructed in intersecting D-brane models,

that is, the starting theory is U(N) SYM.

phenomenology of magnetized brane models
Phenomenology of magnetized brane models

It is important to study phenomenological

aspects of magnetized brane models such as

massless spectra from several gauge groups,

U(N), SO(N), E6, E7, E8, ...

Yukawa couplings and higher order n-point

couplings in 4D effective theory,

their symmetries like flavor symmetries,

Kahler metric, etc.

It is also important to extend such studies

on torus background to other backgrounds

with magnetic fluxes, e.g. orbifold backgrounds.




b個の基底状態    k=0,1,2,…………,(b-1)

2 extra dimensions with magnetic fluxes basic tools
2. Extra dimensions with magnetic fluxes: basic tools

2-1. Magnetized torus model

We start with N=1 super Yang-Mills theory

in D = 4+2n dimensions.

For example, 10D super YM theory

consists of gauge bosons (10D vector)

and adjoint fermions (10D spinor).

We consider 2n-dimensional torus compactification

with magnetic flux background.

We can start with 6D SYM (+ hyper multiplets),

or non-SUSY models (+ matter fields ), similarly.


Higher Dimensional SYM theory with flux

Cremades, Ibanez, Marchesano, ‘04

4D Effective theory <= dimensional reduction

eigenstates of corresponding

internal Dirac/Laplace operator.

The wave functions


Higher Dimensional SYM theory with flux

Abeliangauge field on magnetized torus

Constant magnetic flux

gauge fields of background

The boundary conditions on torus (transformation under torus translations)


Higher Dimensional SYM theory with flux

We now consider a complex field with charge Q ( +/-1 )

Consistency of such transformations under a contractible loop in torus which implies Dirac’s quantization conditions.


Dirac equation on 2D torus

is the two component spinor.

U(1) charge Q=1

with twisted boundary conditions (Q=1)


Dirac equation and chiral fermion

|M| independent zero mode solutions in Dirac equation.

(Theta function)

Properties of theta functions

chiral fermion

By introducing magnetic flux, we can obtain chiral theory.

:Normalizable mode

:Non-normalizable mode


Wave functions

For the case of M=3

Wave function profile on toroidal background

Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.


Fermions in bifundamentals

Breaking the gauge group

(Abelian flux case )

The gaugino fields

gaugino of unbroken gauge

bi-fundamental matter fields

bi fundamental

Gaugino fields in off-diagonal entries

correspond to bi-fundamental matter fields

and the difference M= m-m’ of magnetic

fluxes appears in their Dirac equation.



Zero-modes Dirac equations

No effect due to magnetic flux for adjoint matter fields,

Total number of zero-modes of

:Normalizable mode

:Non-Normalizable mode

4d chiral theory
4D chiral theory

10D spinor

light-cone 8s

even number of minus signs

1st ⇒4D, the other ⇒6D space

If all of appear

in 4D theory, that is non-chiral theory.

If for all torus,


appear for 4D helicity fixed.

⇒4D chiral theory

u 8 sym theory on t6
U(8) SYM theory on T6

Pati-Salam group up to U(1) factors

Three families of matter fields

with many Higgs fields

2 2 wilson lines
2-2. Wilson lines

Cremades, Ibanez, Marchesano, ’04,

Abe, Choi, T.K. Ohki, ‘09

torus without magnetic flux

constant Ai  mass shift

every modes massive

magnetic flux

the number of zero-modes is the same.

the profile: f(y)  f(y +a/M)

with proper b.c.

u 1 a u 1 b theory
U(1)a*U(1)b theory

magnetic flux, Fa=2πM, Fb=0

Wilson line, Aa=0, Ab=C

matter fermions with U(1) charges, (Qa,Qb)

chiral spectrum,

for Qa=0, massive due to nonvanishing WL

when MQa >0, the number of zero-modes

is MQa.

zero-mode profile is shifted depending

on Qb,

pati salam model
Pati-Salam model

Pati-Salam group

WLs along a U(1) in U(4) and a U(1) in U(2)R

=> Standard gauge group up to U(1) factors

U(1)Y is a linear combination.

ps sm
PS => SM

Zero modes corresponding to

three families of matter fields

remain after introducing WLs, but their profiles split



other models
Other models

We can start with 10D SYM,

6D SYM (+ hyper multiplets),

or non-SUSY models (+ matter fields )

with gauge groups,

U(N), SO(N), E6, E7,E8,...

e6 sym theory on t6
E6 SYM theory on T6

Choi, et. al. ‘09

We introduce magnetix flux along U(1) direction,

which breaks E6 -> SO(10)*U(1)

Three families of chiral matter fields 16

We introduce Wilson lines breaking

SO(10) -> SM group.

Three families of quarks and leptons matter fields

with no Higgs fields

splitting zero mode profiles
Splitting zero-mode profiles

Wilson lines do not change the (generation) number of zero-modes, but change localization point.



e8 sym theory on t6
E8 SYM theory on T6

T.K., et. al. ‘10

E8 -> SU(3)*SU(2)*U(1)Y*U(1)1*U(1)2*U(1)3*U(1)4

We introduce magnetic fluxes and Wilson lines

along five U(1)’s.

E8 248 adjoint rep. include various matter fields.

248 = (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + (3,2)(1,-1,-1,1,-1)

+ (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + ......

We have studied systematically the possibilities

for realizing the MSSM.

results e8 sym theory on t6
Results: E8 SYM theory on T6


We can get exactly 3 generations of

quarks and leptons,

but there are tachyonic modes and no top Yukawa.


We allow MSSM + vector-like generations

and require the top Yukawa couplings and

no exotic fields

as well as no vector-like quark doublets.

 three classes

results e8 sym theory on t62
Results: E8 SYM theory on T6

There are many vector-like generations.

They have 3- and 4-point couplings with

singlets (with no hypercharges).

VEVs of singlets  mass terms of vector-like


Light modes depend on such mass terms.

That is, low-energy phenomenologies

depends on VEVs of singlets (moduli).

2 3 orbifold with magnetic flux
2.3 Orbifold with magnetic flux

Abe, T.K., Ohki, ‘08

The number of even and odd zero-modes

We can also embed Z2 into the gauge space.

=> various models, various flavor structures

zero modes on orbifold
Zero-modeson orbifold

Adjoint matter fields are projected by

orbifold projection.

We have degree of freedom to

introduce localized modes on fixed points

like quarks/leptons and higgs fields.

n point couplings and flavor symmetries
N-point couplings and flavor symmetries

The N-point couplings are obtained by

overlap integral of their zero-mode w.f.’s.

zero modes1

Cremades, Ibanez, Marchesano, ‘04

Zero-mode w.f. = gaussian x theta-function

up to normalization factor

products of wave functions
Products of wave functions

products of zero-modes = zero-modes

3 point couplings
3-point couplings

Cremades, Ibanez, Marchesano, ‘04

The 3-point couplings are obtained by

overlap integral of three zero-mode w.f.’s.

up to normalization factor

selection rule
Selection rule

Each zero-mode has a Zg charge,

which is conserved in 3-point couplings.

up to normalization factor

4 point couplings
4-point couplings

Abe, Choi, T.K., Ohki, ‘09

The 4-point couplings are obtained by

overlap integral of four zero-mode w.f.’s.


insert a complete set

up to normalization factor

for K=M+N

n point couplings
N-point couplings

Abe, Choi, T.K., Ohki, ‘09

We can extend this analysis to generic n-point couplings.

N-point couplings = products of 3-point couplings

= products of theta-functions

This behavior is non-trivial. (It’s like CFT.)

Such a behavior wouldbe satisfied

not for generic w.f.’s, but for specific w.f.’s.

However, this behavior could be expected

from T-duality between magnetized

and intersecting D-brane models.

t duality

The 3-point couplings coincide between

magnetized and intersecting D-brane models.

explicit calculation

Cremades, Ibanez, Marchesano, ‘04

Such correspondence can be extended to

4-point and higher order couplings because of

CFT-like behaviors, e.g.,

Abe, Choi, T.K., Ohki, ‘09

non abelian discrete flavor symmetry
Non-Abelian discrete flavor symmetry

The coupling selection rule is controlled by

Zg charges.

For M=g, 1 2 g

Effective field theory also has a cyclic permutation symmetry of g zero-modes.

These lead to non-Abelian flavor symmetires

such as D4 and Δ(27)Abe, Choi, T.K, Ohki, ‘09

Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04

T.K. Nilles, Ploger, Raby, Ratz, ‘06

permutation symmetry d brane models
Permutation symmetry D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

There is a Z2 permutation symmetry.

The full symmetry is D4.

permutation symmetry d brane models1
Permutation symmetry D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

geometrical symm. Full symm.

Z3 Δ(27)

S3 Δ(54)

intersecting magnetized d brane models
intersecting/magnetized D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

generic intersecting number g

magnetic flux

flavor symmetry is a closed algebra of

two Zg’s.

and Zg permutation

Certain case: Zg permutation larger symm. Like Dg

magnetized brane models
Magnetized brane-models

Magnetic flux M D4

2 2

4 1++ + 1+- +1-+ + 1--

  ・・・           ・・・・・・・・・

Magnetic flux M Δ(27) (Δ(54))

3 31

6 2 x 31

9 ∑1n n=1,…,9

(11+∑2n n=1,…,4)

 ・・・           ・・・・・・・・・

non abelian discrete flavor symm
Non-Abelian discrete flavor symm.

Recently, in field-theoretical model building,

several types of discrete flavor symmetries have

been proposed with showing interesting results,

e.g. S3, D4, A4, S4, Q6, Δ(27), ......

Review: e.g

Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10

⇒ large mixing angles

one Ansatz: tri-bimaximal

applications of couplings
Applications of couplings

We can obtain quark/lepton masses and mixing angles.

Yukawa couplings depend on volume moduli,

complex structure moduli and Wilson lines.

By tuning those values, we can obtain semi-realistic results.

Abe, Choi, T.K., Ohki ‘08

T.K., et. al. ‘10,

Abe, et. al. work in progress

Other applications

4 other studies superspace discription
4. Other studiesSuperspacediscription

Abe, T.K. , Ohki, Sumita, ’ 12

Superspace formulation for extra-dimentional

field theory with magnetic fluxes

including moduli.

Off-shell formalism

⇒  modulus mediation

massive modes
Massive modes

We can compute mass spectra of massive KK modes

and their wavefunctions.

Then, we can compute couplings among

massless and massive modes.

Hamada, et. al. work in progress

Interesting applications


We have studiedphenomenological aspects

of magnetized brane models.

Model building from U(N), E6, E7, E8

N-point couplings are comupted.

4D effective field theory has non-Abelian flavor

symmetries, e.g. D4, Δ(27).

Orbifold background with magnetic flux is

also important.

further studies
Further studies

Derivation of realistic values of quark/lepton

masses and mixing angles.

Applications of flavor symmetries

and studies on their anomalies

Phenomenological aspects of massive modes