Yet Another Alternative to Compactification

Transcription

1 Okayama Institute for Quantum Physics: June 26, 2009 Yet Another Alternative to Compactification Heterotic five-branes explain why three generations in Nature arxiv: [hep-th] Tetsuji KIMURA (KEK) with Shun ya Mizoguchi (KEK, SOKENDAI)

2 Introduction

3 Three Generations in Nature Necessary condition of CP-violation: Kobayashi-Maskawa No sufficient conditions in the Standard Model Tried to embed this problem in GUTs and String Theories TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 3

4 Particles in the Standard Model quark lepton gauge field Higgs ( ul d L particles ) ( ) ( ) ul ul d L d L quantum number spin SU(3) C SU(2) W U(1) Y 3 2 1/6 u R u R u R 3 1 2/3 d R d R d R 3 1 1/3 ( ) νel 1 2 1/2 e L e R ν er A a µ W µ ± Wµ /2 1/2 1 B µ ( ) h /2 0 h (only one generation in fermions) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 4

5 SU(5) Grand Unified Model d c R d c R d c R ν el e L 0 u c R u c R u L d L u c R 0 u c R u L d L u c R u c R 0 u L d L u L u L u L 0 e c R d L d L d L e c R 0 ν e R U(1) Y : SU(5) 1 2 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 5

6 Further GUTs SU(5) matter Higgs SO(10) E 6 27 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 6

7 A typical example: Calabi-Yau compactification in E 8 E 8 heterotic string theory Standard Embedding: ω m ab A ab m SU(3) E 8 E 6 SU(3) # of generations = 1 2 χ(cy) = h 1,1 h 2,1 h 1,1 = # of Kähler moduli = # of (27, 3) repr. (size of CY) h 2,1 = # of complex structure moduli = # of (27, 3) repr. (shape of CY) M.B. Green, J.H. Schwarz and E. Witten: Chapter 16.2 Problems So many Calabi-Yau manifolds So many massless modes appear in four-dimensional physics flux compactification to yield potential energy TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 7

8 Another example: D-brane scenario in higher-dimensional theories ex.) N f D7-branes (green planes) N c D3-branes (red lines) SU(N c ) gauge theory with N f flavors Good Points Simple and visible Problems a bit artificial setup TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 8

9 Yet Another Alternative: Intersecting 5-branes in hetertotic string theory Our Model brane 5-brane our world Good Points Simple! E 6 gauge symmetry appears Naturally obtain 3 E 6 -charged multiplets in four dimensions as Nambu-Goldstone modes (different counting of E 6 -charged matter fields from CY) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 9

10 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions

11 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions

12 Heterotic String Theory in Ten Dimensions string Objects: coupled to B-field electrically/magnetically NS5-brane 16 Supersymmetry Charges E 8 E 8 or SO(32) gauge symmetries Effective action in the string frame is given as S boson = 1 2κ 2 d 10 x { g e 2φ R + 4( M φ) 2 1 } 10 3 H2 MNP α 30 TrF MN TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 12

13 Supersymmetry transformations: gravitino δψ M = ( M ω M AB Γ AB ) ɛ gaugino dilatino δλ = 1 4 δχ = 1 4 F MNΓ MN ɛ ( Γ M M φ 1 6 H MNP Γ MNP ) ɛ Bianchi identity (via anomaly cancellation) [ ] 1 dh = α 30 Tr(F F ) tr(r(ω +) R(ω + )) with ω ±M AB := ω M AB ± H M AB TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 13

14 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions

15 A Five-brane Solution: Symmetric solution ds 2 = 5 µ,ν=0 η µν dx µ dx ν + e 2φ A m = 2ρ 2 N Σ mn x n e 2φ = e 2φ 0 + Nα x 2, x2 := 9 m,n=6 δ mn dx m dx n x 2 (x 2 + ρ 2 N ), ρ2 N := Nα e 2φ, Σ mn := ɛ mn pq Σ pq 9 (x m ) 2 m=6 H mnp = ɛ mnp q q φ ω +m ab = ω m ab + H m ab = 2σ mn ab n φ σ mn ab := δ mn ab 1 2 ɛ mn ab, σ mn ab = ɛ mn pq σ pq ab C.G. Callan, J.A. Harvey and A. Strominger, Nucl.Phys.B 359 (1991) 611 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 15

16 8 SUSY preserved (8 broken) NS5-brane A m : SU(2) self-dual instanton in 4 transverse directions ω +m : SU(2) self-dual We can embed ω +m into A m SU(2) spin SU(2) gauge =: SU(2) frozen E 8 gauge symmetry is broken to E 7 SU(2) frozen 120 bosonic moduli = 30 hypermultiplets in 6-dim l theory 28 SU(2)-Nambu-Goldstone hypermultiplets TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 16

17 Counting hypermultiplets in 6-dim l theory (exercise to E 6 ) E 8 E 7 SU(2) 248 = (133, 1) (1, 3) (56, 2) }{{} SU(2)-Nambu-Goldstone SU(2) instanton moduli in 4-transverse space: 4 translations 1 scale of instanton = = /4 = 30 hypermultiplets in 6-dim l theory ((56 2)/4 = 28 of them from SU(2)-Nambu-Goldstone) C.G. Callan, J.A. Harvey and A. Strominger, Nucl.Phys.B 367 (1991) 60 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 17

18 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions

19 Intersection Rule for p-branes R. Argurio, F. Englert and L. Houart, Phys.Lett.B 398 (1997) 61 q + 1 = (p A + 1)(p B + 1) D (ε Aa A )(ε B a B ) D : q : p A : total spacetime dimensions intersecting dimensions spatial dimensions of p A -brane a A : 1 (NSNS B-field), 1 2 (5 p A) (RR (p A + 1)-form) ε A : +1 (electric), 1 (magnetic) Now, we consider two intersecting NS5-branes in heterotic string: q + 1 = (5 + 1)(5 + 1) ( 1)( 1)( 1)( 1) q = 3 good dimensions to consider four-dimensional spacetime TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 19

20 brane 5-brane our world The string frame metric and the solution are given as ds 2 = η µν dx µ dx ν + h (dx m ) 2 + h 2 (dx m ) 2 µ,ν=0 m=4 m=8 h = 1 + N x 8, h 2 = e 2φ, H 459 = H 679 = N 2 x8 K. Ohta and T. Yokono, JHEP 02 (2000) our new idea TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 20

21 Spin connections (ω ± ) m ab are expressed as (ω ± ) 4 = N x8 2h ± C A, (ω ±) 5 = N x8 2h ±1 1 1 C A (ω ± ) 6 = N x8 2h ± C A, (ω ±) 7 = N x8 2h 3 2 (ω ± ) 8 = 0, (ω ± ) 9 = N x8 2h ±1 ±1 1 1 ±1 1 C A 1 C A Each spin connection belongs to SU(3) group However, ω + does not correspond to ω by any similarity transformations Embedding ω +m A m, the Bianchi identity is given by [ ] 1 dh = α 30 Tr(F F ) tr(r(ω +) R(ω + )) = 0 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 21

22 4 SUSY preserved (12 broken) ω +m : an SU(3) holonomy connection in 6 directions ω m : another SU(3) holonomy connection in 6 directions We can embed ω +m into A m SU(3) spin SU(3) gauge =: SU(3) frozen E 8 gauge symmetry is broken to E 6 SU(3) frozen 4-dim l N = 1 E 6 on the intersecting spacetime 3 E 6 -charged complex matter multiplets = SU(3)-Nambu-Goldstone modes TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 22

23 Counting E 6 -charged chiral multiplets in 4-dim l theory E 8 E 6 SU(3) 248 = (78, 1) (1, 8) (27, 3) (27, 3) }{{} SU(3)-Nambu-Goldstone Focus only on SU(3)-Nambu-Goldstone E 6 fundamental: (27 3) + (27 3) (27, 3) and (27, 3) are complex conjugate {(27 3) + (27 3)}/2 = 27 3 SU(3)-Nambu-Goldstone complex chiral bosons = 3 copies of complex chiral bosons of E 6 fundamental repr. cf) C.W. Bernard, N.H. Christ, A.H. Guth and E.J. Weinberg, Phys. Rev. D16 (1977) 2967 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 23

24 Fermionic moduli via broken SUSY parameters: Generically, 6-dim l space has an SO(6) SU(4) holonomy group 1/4 SUSY yields 1 Killing spinor ɛ 4 and 3 broken spinors ɛ i in 6 dimensions (x 4,..., x 9 ) δψ m = ( m ω m ab γ ab ) ɛ = } {{ } SU(3) holonomy ɛ 1 ɛ 2 ɛ 3 ɛ 4 = 0 Furthermore, we embed ω +m A m SU(3) spin SU(3) gauge # of broken spinors = # of SU(3) gauge non-singlet fermions in 4 dimensions They are SU(3)-Nambu-Goldstone fermions in 4-dim l spacetime and behave as 3 copies of complex moduli of 27 under E 6 transformations TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 24

25 The SU(3)-Nambu-Goldstone bosons and fermions are combined into 3 copies of complex chiral multiplets of E 6 fundamental repr.! 3 copies = 3 generations! Remark This intersecting five-brane configuration is T-dual to deformed conifold (non-compact CY with h 2,1 = 1, h 1,1 = 0) # of generations is just one!? (from ordinary viewpoint) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 25

26 Different counting of generations from CY compactification We counted broken SUSY parameters ɛ i, which are three times as many as the Killing spinor ɛ 4. Dirac index is sensitive to # of set of (27, 3), but insensitive to the way how these 27 are embedded into the original E 8 repr. possibility of multiplication three by the Dirac index! Then, the factor three differs from the counting in the CY compactification deformed conifold (# is one) intersecting five-brane (# is three) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 26

27 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions

28 Yet Another Alternative to Compactification (cf. Randall-Sundrum 1 model) Compactify all extra directions (x 4,..., x 9 ) to six-torus (T 5 (S 1 /Z 2 )) h x h Π h 0 2Πr c Πr c Πr c 2Πr c x Notice: In order to introduce objects which absorb/emit NS charges, we have to put another intersecting 5-branes with negative tension in the x 8 direction Then, we modify the function h to h(x 8 ) = h 0 + N x 8 2πkr c, k Z TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 28

29 h x h Π h 0 2Πr c Πr c Πr c 2Πr c x This setup has the following features: Cosmological constant vanishes Warp factor is milder (linear) than that of RS1 (exponential) Supersymmetry is broken completely! We obtain 4-dim l non-susy model with E 6 gauge symmetry and 3 generations! (under the vanishing limit of h 0 ) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 29

30 Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Model Summary and Discussions

31 Summary Studied NS5-brane and its intersection in heterotic string Obtained a simple model to yield 3 generations in four dimensions Applied it to consider a non-susy model via torus compactification Unified Nambu and Kobayashi-Maskawa Nobel Prize in Physics 2008 Discussions Connecting to bottom-up model-building (cf. Maekawa et al.) More understanding intersecting NS5-branes Comparison to type II and F-theoretical configurations via string dualities

32 Thank You