1. Three family GUT-like models from heterotic string
Based on arXiv: [hep-th]
Kei-Jiro Takahashi (Kyoto Univ.)

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 approaches are proposed so far:
Heterotic string → Calabi-Yau, orbifold…
Type II string → Intersecting D-brane, flux,…
GUT and SM gauge groups are naturally included in E8.
Internal space
SU(3)xSU(2)xU(1) E8 E6
E5=SO(10)
E4=SU(5)

3. A standard-like story to the Standard Model
scale
String theory 18
10 GeV
TypeIIA,B, Heterotic
UV completion
mass 0
1×Mst
2×Mst
3×Mst　・・・
Massive states are decoupled in low energy scale.
10D SUGRA
N=1 YM
N=1 Gravity
compactification
matter
6~?
10 GeV
15-16
10 GeV
GUT breaking
moduli
Hidden sector
By gaugino condensation,
SUSY breaking may occur.
3~?
SUSY breaking
mediation
10 GeV
SUSY SM m
SUSY partner mu md ms mb mt mc mh me mμ mτ mν 2
U(1)
SU(2)
SU(3)
10 GeV
Standard model
Gravity

4. Why orbifold? N=1 supersymmetry:
Torus compactification gives N=4 model.
N=2 and 4 is not chiral.
Requirement of naturalness (for higgs mass), but is not necessary.
Stability of compact space (no tadpole, no tachyon).
Appearance of “matter”:
Geometry of compact space generate localized string states, which are not states obtained from dimensional reduction.
I think orbifold provides good playground for model construction.

5. Torus and symmetry
The rotations of an orbifold, i.e. θ and φ, should be given by the symmetry of the torus.
We generalize these concepts to six dimensional torus.
Examples in 2D :
T2/Z3
T2/Z2
fixed point

6. String on orbifolds: Geometry and structure → spectrum
An example of Z3 orbifold on factorizable torus:
In this description, an interaction of particles could be interpreted as a change of the localizing points in the compact space.
3x3x3=27
twisted sectors +
9 untwisted sectors
36 generations of matter
Boundary cond. X(2π)=θX(0)+ v , θ: rotation, v: shift (v ∈Λ) d
d + H → u + u + H
W, Z..

7. E6 torus θ-sector ⇒ = 72 E6 root lattice: Orbifold action of Z3xZ3:
3 fixed tori appear in θ-twisted sector !
θφ -sector includes 27 fixed points.
rotation
θ-sector　⇒
This is similar for φ, θφ-sectors.
= 72 2

8. A model with the standard embedding
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 + N=1 Gravity + N=1 YM
half of χ

9. An SU(5) model Quite simple assumptions:
Z3xZ3 orbifold on E6 root lattice
+ gauge embeddings:
Gauge group:
No adjoint higgs! ← Not realistic.
Assume the vector-like 6’s of SU(6)’ have large mass (~MGUT), the coupling of SU(6)’ become strong at ~ 10 GeV.
Higgs
3-family matter
visible
hidden
Messenger 9

10. An SO(10) model Similarly with the gauge embeddings: Gauge group:
No adjoint Higgs.
visible
hidden
Higgs
3-family matter
Messenger

11. The allowed 3-point functions
There are mainly three requirements for the allowed couplings.
Gauge invariance (charge conservation).
H-momentum (10D angular momentum ) conservation.
Geometric restrictions (instanton effect).

12. A little phenomenology of the SO(10) model
Assuming that all the pairs of 7 and 7 generate mass terms and decouple, the hidden sector scale is
Because C (1,2) couples only to C (1,2)(7) and U2(7), the tree level superpotential includes W= WADS + C C U SUSY ?
Hidden sector

13. Summary
The origin of three generation matter are explained by the three fixed tori in a twisted sector in the models.
Due to small number of fixed tori, the spectra are very simple.
In Z3xZ3 orbifold, three point (Yukawa) interactions with flavor mixing terms, which are suppressed by e , are generated by instanton effect.
They include strongly coupled sector in the low energy,
and may cause spontaneous SUSY breaking.
Nevertheless they are toy models without adjoint higgs.
By inclusion of Wilson lines, we can obtain directly SU(3)xSU(2)xU(1) gauge group. → More realistic models?
String theory indicates various symmetries or textures in the low energy spectra. Model constructions including many phenomenological features (realization of the spectrum, SUSYV, its mediation, moduli stabilization, etc) will be more exciting in the future. -s