Heterotic orbifold models on E 6 Lie root lattice

1. Based on arXiv: 0707.3355 (ver.2). Kei-Jiro Takahashi(KyotoUniv.) Heterotic orbifold modelson E6 Lie root lattice SI2007 @ Fuji-yoshida

2. Introduction • To realize the Standard Model in low energy, we consider compactification of E8xE8 heterotic string on 6D orbifold. • 4D + compact 6D (orbifold) → N=1 spectrum • We develop orbifolds on “generic tori”, i.e. non-factorizable. All the massless states of this model. An SO(10)model Higgs 3-family + vector-like Messenger??

3. mc mb mt me mμ mτ mν mν mν mediation m m m SUSY breaking SUSY partner mu md ms U(1) SU(2) SU(3) mh A standard-like story to the Standard Model scale String theory 18 10GeV 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~? 10GeV 4D SUGRA moduli Hidden sector By gaugino condensation, SUSY breaking may occur. 3~? SUSY SM 10GeV 2 Standard model Gravity 10GeV

4. 6 6 6 T = R / Λ 6 O = T / P Orbifold (Dixon, Harvey, Vafa, Witten NPB261(85)678, 274(86)285) • We can obtain an orbifold to identify the points on torus by rotationand shift. • Examples for Z2 orbifold on SO(4) and SU(3) lattices. Λ: compactification lattice P: point group (rotation) Boundary cond. X(2π) = θX(0) + v , θ: rotation, v: shift (v ∈Λ) 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)

5. → graviton, gauge, matter,… → matter,… Orbifold and heterotic string • Orbifold in 6D compact space An example of Z3 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 Untwisted Twisted

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 u + H 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)xSU(3)xSU(3), G2xSU(3)xSO(4)… → factorizable SO(6)xSO(6), SO(8)xSO(4), SU(7), SO(12), E6 …→ non-factorizable SU(3) G2 ～ SU(4) SO(6) ～ SO(12) E6 (K.T with T.Kimura, M.Ohta, To appear some time.)

8. Point group elements of orbifolds (K.T. JHEP03(07)103) • 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 ZnxZm orbifold action. • The rank of the groups: e.g. general Z2 elements rank

9. E6 torus • E6 root lattice: • Orbifold action of Z3xZ3: • 3 fixed tori appear in θ-twisted sector ! • θΦ -sector includes 27 fixed points (and has some complexity). rotation θ-sector　⇒ This is similar for Φ, θΦ-sectors. 2

10. 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: + singlets +N=1 Gravity+N=1 YM 36generations 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: Z3xZ3 orbifold on E6 root lattice + gauge embeddings: • Gauge group: • Assume the vector-like (7)s have large mass (~MGUT), the coupling of SU(7)’ become strong at ~ 10 GeV. visible hidden Higgs 3-family matter Messenger?? 7

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: visible hidden Higgs 3-family matter

13. Summary • We first proposed heterotic orbifold construction on E6 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.