1. Heterotic orbifold models on E 6 Lie root lattice SI2007 @ Fuji-yoshida Based on arXiv: 0707.3355 (ver.2). Kei-Jiro Takahashi (Kyoto Univ.)

2. Introduction  To realize the Standard Model in low energy, we consider compactification of E 8 xE 8 heterotic string on 6D orbifold.  4D + compact 6D (orbifold) → N=1 spectrum  We develop orbifolds on “generic tori”, i.e. non-factorizable. Messenger?? Higgs An SO(10) model 3-family + vector-like All the massless states of this model.

3. A standard-like story to the Standard Model String theory 10D SUGRA 4D SUGRA mass 01 × M st 2 × M st 3 × M st ・・・ compactification N=1 GravityN=1 YM moduli UV completion Massive states are decoupled in low energy scale. scale 10 GeV 18 SUSY SM Standard model Hidden sector By gaugino condensation, SUSY breaking may occur. 10 GeV 3~? 10 GeV 2 6~? Gravity TypeIIA,B, Heterotic matter U(1) SU(2)SU(3) mumu mdmd msms mbmb mtmt mcmc mhmh meme mμmμ mτmτ mνmν mνmν mνmν mmm SUSY partner SUSY breaking mediation

4. Orbifold  We can obtain an orbifold to identify the points on torus by rotation and shift.  Examples for Z 2 orbifold on SO(4) and SU(3) lattices. T = R / Λ 6 66 O = T / P 6 P : point group (rotation) Λ : compactification lattice Boundary cond. X(2π) = θX(0) + v, θ : rotation, v : shift (v ∈ Λ) A fixed line by Z 2. Action of Z 2 : (Dixon, Harvey, Vafa, Witten NPB261(85)678, 274(86)285) (Z 2x Z 2 non-factorizable: A.Faraggi et.al hep-th/0605117, S.Forste, T.Kobayashi, H.Ohki, K.T hep-th/0612044)

5. Orbifold and heterotic string  Orbifold in 6D compact space An example of Z 3 orbifold  Twisted sectors are necessary to satisfy the modular invariance of string one-loop amplitude. Thus we should count all modes which are allowed by the boundary conditions. 3x3x3= 27 twisted sectors + 9 untwisted sectors 36 generations of matter Mode expansions → graviton, gauge, matter,… → matter,… Twisted Untwisted

6. Geometry and structure → spectrum  In the heterotic string picture, the locations in the compact space corresponds to the flavors of matter. (This figure depicts only the concepts of geometry and the spectrum. Realistic models are more complex, and not easy like this.) d + H → u d H u + + W, Z..

7. Lie root lattices ⊃ 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) x SU(3) x SU(3), G 2x SU(3) x SO(4)… → factorizable SO(6) x SO(6), SO(8) x SO(4), SU(7), SO(12), E 6 …→ non-factorizable ～ SU(3) SU(4)SO(6) SO(12) E6E6 ～ G2G2 (K.T with T.Kimura, M.Ohta, To appear some time.)

8. Point group elements of orbifolds rank e.g. general Z 2 elements  The elements of orbifold should act crystallographically on torus, i.e. symmetry of the lattice. ・ Weyl reflection ・ Graph automorphism (outer automorphism)  We select two commutative elements as Z n xZ m orbifold action.  The rank of the groups: (K.T. JHEP03(07)103)

9. E 6 torus θ-sector ⇒ This is similar for Φ, θΦ-sectors. 2  E 6 root lattice:  Orbifold action of Z 3 xZ 3 :  3 fixed tori appear in θ-twisted sector !  θΦ -sector includes 27 fixed points (and has some complexity). rotation

10. E 6 model for the standard embeddings  An explicit model on Z 3 x Z 3 orbifold on E 6 lattice. To satisfy the modular invariance we must embed the twist in 6D space to E 8 xE 8 gauge space.  Gauge group:  The massless spectrum: 36 generations+ singlets + N=1 Gravity + N=1 YM With corrections in my paper of ver.2. Thanks to F.Ploeger, S.Ramos-Sanchez and P.Vaudrevange.

11. An SO(10) model  Quite simple assumptions: Z 3 xZ 3 orbifold on E 6 root lattice + gauge embeddings:  Gauge group:  Assume the vector-like (7)s have large mass (~M GUT ), the coupling of SU(7)’ become strong at ~ 10 GeV. visible hidden Messenger?? Higgs 7 3-family matter

12. An SU(5) model  Similarly + gauge embeddings:  Gauge group:  For GUT breaking if one of (5,1,1) has VEV, the gauge group break to Pati-Salam type: 3-family matter Higgs visible hidden

13. Summary  We first proposed heterotic orbifold construction on E 6 root lattice and obtain very simple GUT-like models from heterotic orbifolds.  We can add Wilson lines to this construction, and will lead to other interesting models.  It may contain SUSY hidden sector (J.E.Kim, arXiv:07060293) and its messenger.  In string theory different models are related by dualities and some symmetries. Such interesting coincidence are pointed out in some cases (M.Ratz, et.al. hep-th/0702176). Then it is valuable for itself to investigate geometries of compact space.  From UV to IR, it is challenging to construct more realistic model.