KeiJiro Takahashi Kyoto Univ'

1
Three-family heterotic orbifold models on E6
root lattice

• Kei-Jiro Takahashi (Kyoto Univ.)
• arXiv0707.3355

2
Introduction

• How can we realize the Standard Model as a low
  energy effective theory of string theory?
• As a perturbative approach, we assume the
  compactification
• 4D 6D compact space
• Several ways are proposed so far
• Heterotic string ? Calabi-Yau,
• 
  orbifold,
• 
  fermionic construction,
• Type II string ? Intersecting D-brane,
  flux,
• We construct new (rather missed) orbifold models
  based on E8xE8 heterotic string.

Internal space
3
A standard-like story to the Standard Model
scale
String theory
18
10 GeV
TypeIIA,B, Heterotic
UV completion
mass 0
1Mst
2Mst
3Mst
Massive states are decoupled in low energy scale.
10D SUGRA
N1 Gravity
N1 YM
compactification
matter
6?
10 GeV
4D SUGRA
moduli
Hidden sector
By gaugino condensation, SUSY breaking may occur.
3?
SUSY SM
10 GeV
2
Standard model
Gravity
10 GeV
4
Orbifold (as compact space )

• We can obtain an orbifold to identify the points
  on torus by rotation and shift.
• Examples for Z2 orbifold on SO(4) and SU(3)
  lattices.

? compactification lattice
P point group (rotation)
Boundary cond. X(2p) ?X(0) v , ?
rotation, v shift (v ??)
Action of Z2
A fixed line by Z2
(Z2xZ2non-factorizable A.Faraggi et.al
hep-th/0605117, S.Forste, T.Kobayashi, H.Ohki,
K.T hep-th/0612044)
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Geometry and structure ? spectrum

• An example of Z3 orbifold on factorizable torus
• In this description, the interaction of particles
  could be interpreted as change of the loci in the
  compact space.

3x3x327 twisted sectors

9 untwisted sectors
36 generations of matter

d H ? u
d
u

H
W, Z..
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Non-factorizable lattices

• We define the shape of torus by the words of Lie
  algebra.
• We take direct products of these tori,
• and the compact space should be totally six
  dimensions.
• SU(3)xSU(3)xSU(3), G2xSU(3)xSO(4) ?
  factorizable
• SU(4)xSU(4), SO(8)xSO(4), SU(7), SO(12), E6 ?
  non-factorizable

SU(3)
G2
SO(12)

SU(4)
SO(6)
E6

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Point group elements of orbifold

• The elements of orbifold should act
  crystallographically on torus, i.e. symmetry of
  the lattice.
• Weyl reflection
• Graph automorphism (outer automorphism)
• For ZnxZm orbifold, the classification by
• the coxeter elements is incomplete.
• We select two commutative elements
• as ZnxZm orbifold action.

(K.T. JHEP03(07)103)
(K.T with T.Kimura, M.Ohta)
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E6 torus

• E6 root lattice
• Orbifold action of Z3xZ3
• 3 fixed tori appear in ?-twisted sector !
• ?F -sector includes 27 fixed points.

rotation
?-sector ?
This is similar for F, ?F-sectors.
2
9
E6 model for the standard embeddings

• An explicit model on Z3xZ3 orbifold on E6
  lattice.
• To satisfy the modular invariance we must embed
  
• the twist in 6D space to E8xE8 gauge space.
• Gauge group
• The massless spectrum

36 generations
singlets N1 Gravity N1 YM
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An SO(10) model

• Quite simple assumptions
• Z3xZ3 orbifold on E6 root lattice
• gauge embeddings
• Gauge group
• No adjoint higgs! ? Not realistic.
• Assume the vector-like (7)s have large mass
  (MGUT), the coupling of SU(7) become strong at
  10 GeV.

visible
3-family matter
hidden
Messenger??
7
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An SU(5) model

• Similarly
• gauge embeddings
• Gauge group

visible
hidden
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The advantages of these models

• The origin of three generation of matter are
  explained by the three fixed tori in a twisted
  sector.
• In Z3xZ3 orbifold, three point (Yukawa)
  interactions , which are suppressed by e , are
  generated by instanton effect.
• They include strong coupling sector in the low
  energy,
• and may cause spontaneous SUSY breaking.
• Nevertheless they are toy models without adjoint
  higgs.

-S
Hidden sector
strong
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Summary

• We first construct heterotic orbifolds on E6 root
  lattice and obtain very simple GUT-like models.
  These models have some favorable features for
  phenomenology.
• In order to realize directly SU(3)xSU(2)xU(1)
  models, we can add Wilson lines to this
  construction, and it may lead to a more realistic
  model.
• In string theory different models are related by
  dualities and symmetries. Some coincidence
  between non-factorizable models and factorizable
  models with generalized discrete torsion are
  pointed out in hep-th/702176. It is valuable to
  investigate geometries of compact space by
  itself.
• From UV to IR, it is a challenging issue to
  construct more realistic model, that is
  perturbative vacua of string theory.