Okayama Institute for Quantum Physics: June 26, 2009 Yet Another Alternative to Compactification Heterotic five-branes explain why three generations in Nature arxiv: 0905.2185 [hep-th] Tetsuji KIMURA (KEK) with Shun ya Mizoguchi (KEK, SOKENDAI)
Introduction
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
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 1 1 1 ν er 1 1 0 A a µ 8 1 0 W µ ± Wµ 3 1 3 0 1/2 1/2 1 B µ 1 1 0 ( ) h 0 1 2 1/2 0 h (only one generation in fermions) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 4
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 5 10 1 U(1) Y : + 1 3 + 1 3 + 1 3 1 2 SU(5) 1 2 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 5
Further GUTs SU(5) matter 5 10 1 Higgs 5 5 1 SO(10) 16 10 1 E 6 27 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 6
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
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
Yet Another Alternative: Intersecting 5-branes in hetertotic string theory Our Model 0 1 2 3 4 5 6 7 8 9 5-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
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions
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 2 +... TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 12
Supersymmetry transformations: gravitino δψ M = ( M + 1 4 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
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions
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 := + 1 2 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 = + 1 2 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
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
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 56 2 + 1 3 = 115 4 + 1 + 115 = 120 120/4 = 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
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions
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 2 1 2 (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) 10 2 1 2 ( 1)( 1)( 1)( 1) q = 3 good dimensions to consider four-dimensional spacetime TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 19
0 1 2 3 4 5 6 7 8 9 5-brane 5-brane our world The string frame metric and the solution are given as ds 2 = 3 7 9 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) 023 + our new idea TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 20
Spin connections (ω ± ) m ab are expressed as (ω ± ) 4 = N x8 2h 3 2 0 B @ +1 ±1 1 1 1 C A, (ω ±) 5 = N x8 2h 3 2 0 B @ +1 1 ±1 1 1 C A (ω ± ) 6 = N x8 2h 3 2 0 B @ +1 ±1 1 1 1 C A, (ω ±) 7 = N x8 2h 3 2 (ω ± ) 8 = 0, (ω ± ) 9 = N x8 2h 0 B @ 0 B @ +1 +2 1 2 1 ±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
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
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
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 + 1 4 ω m ab γ ab ) ɛ = 0 0 0 0 0 0 0 0 } {{ } 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
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
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
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Alternative to Compactification Summary and Discussions
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
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
Contents Introduction Heterotic String Theory Five-brane Solutions Intersection Yet Another Model Summary and Discussions
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
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