Yet Another Alternative to Compactification by Heterotic Five-branes

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1 The University of Tokyo, Hongo: October 26, 2009 Yet Another Alternative to Compactification by Heterotic Five-branes arxiv: [hep-th] Tetsuji KIMURA (KEK) Shun ya Mizoguchi (KEK, SOKENDAI)

2 Introduction

3 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 = 2 χ(cy) = h, h 2, h, = # of Kähler moduli = # of (27, 3) repr. (size of CY) h 2, = # of complex structure moduli = # of (27, 3) repr. (shape of CY) M.B. Green, J.H. Schwarz and E. Witten: Chapter 6.2 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 3

4 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 4

5 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 E 6 -charged multiplets in four dimensions as Nambu-Goldstone modes TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 5

6 Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions

7 Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions

8 Heterotic String Theory in Ten Dimensions string Objects: coupled to B-field electrically/magnetically NS5-brane 6 Supersymmetry Charges E 8 E 8 or SO(32) gauge symmetries Effective action in the string frame is given as S boson = 2κ 2 d 0 x { g e 2φ R + 4( M φ) 2 } 0 3 H2 MNP α 30 TrF MN TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 8

9 Supersymmetry transformations: gravitino δψ M = ( M + 4 ω M AB Γ AB ) ɛ gaugino dilatino δλ = 4 δχ = 4 F MNΓ MN ɛ ( Γ M M φ 6 H MNP Γ MNP ) ɛ Bianchi identity (via anomaly cancellation) [ ] 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 9

10 Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions

11 Intersection Rule for p-branes R. Argurio, F. Englert and L. Houart, Phys.Lett.B 398 (997) 6 q + = (p A + )(p B + ) D 2 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 : (NSNS B-field), 2 (5 p A) (RR (p A + )-form) ε A : + (electric), (magnetic) Now, we consider two intersecting NS5-branes in heterotic string: q + = (5 + )(5 + ) ( )( )( )( ) q = 3 good dimensions to consider four-dimensional spacetime TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION

12 brane 5-brane our world The string frame metric and the solution are given as ds 2 = 2 6 η µν dx µ dx ν + h 2 (dx m ) 2 + h (dx m ) 2 µ,ν=0,7,8,9 m= m=3 h = + N x = e φ, H 234 = H 256 = N 2h x K. Ohta and T. Yokono, JHEP 02 (2000) our new idea TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 2

13 Spin connections (ω ± ) m a b (ω ± H) m a b are expressed as (ω ± ) = 0, (ω ± ) 2 = h (ω ± ) 3 = h 2h 3 2 (ω ± ) 5 = h 2h ± ± 2h C A, (ω ±) 4 = h 2h 3 2 C A, (ω ±) 6 = h 2h ± ± ± ± C A C A 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 [ ] dh = α 30 Tr(F F ) tr(r(ω +) R(ω + )) = 0 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 3

14 4 SUSY preserved (2 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 = with E 6 -gauge symmetry on the intersecting spacetime TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 4

15 Counting E 6 -charged chiral multiplets in 4-dim l theory E 8 E 6 SU(3) 248 = (78, ) (, 8) (27, 3) (27, 3) Focus only on E 6 fundamental: (27 3) + (27 3) (27, 3) and (27, 3) are complex conjugate {(27 3) + (27 3)}/2 = 27 3 complex bosons = 3 complex bosons of E 6 fundamental repr. cf) C.W. Bernard, N.H. Christ, A.H. Guth and E.J. Weinberg, Phys. Rev. D6 (977) 2967 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 5

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

17 The Answer: one generation See the Dirac equation of gaugino in 0D 0 = /D(ω 3 H, A)χ ΓM χ M Φ + 4 ΓM Γ AB F AB (ψ M + 2 ) 3 Γ Mλ with background ψ M = 0 = λ and χ = e Φ χ: 0 = /D(ω 3H, A)χ 0 = Γ µ µ χ + Γ m D m (ω 3H, A)χ Focus only on the fermionic modes χ belonging to (27, 3) or (27, 3). Factorize χ = χ 4D ψ 6D and evaluate the second term as the mass term of χ 4D. TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 7

18 Assume that χ depends only on x as a smeared configuration (cf. the metric) 0 = Γ a m e ( a x mχ (x ) + Γ a i ) e a 4 (ω 3 H) m ab Γ ab + A m χ with where e a m = h h h 2 h 2 h 2 h 2, h = + N x Γ = γ # γ, Γ 2 = γ # γ 2, Γ 3 = γ # γ 3, Γ 4 = γ # γ 4, Γ 5 = γ # γ 5, Γ 6 = γ # γ 6 γ # : chirality operator in 4D, γ a : gamma matrix in 6D γ = σ γ 2 = σ σ 2 γ 3 = σ σ 2 2 γ 4 = σ σ 3 σ γ 5 = σ σ 3 σ 2 γ 6 = σ σ 3 σ 3 TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 8

19 Spin connections (ω 3 H) m a b ω m a b are expressed as bω = 0, bω 2 = h bω 3 = h 2h h C A, bω 4 = h 2h C A C A bω 5 = h 2h C A, bω 6 = h 2h C A h = h/ x = x : step function TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 9

20 Dirac equation of 6D part is given as 0 = γ e d ( dx ψ 6D + γ a i ) e a 4 (ω 3 H) m ab γ ab + A m ψ 6D ( ) ( 0 ih 2 2 d 0 M ih dx ψ 6D + M 2 0 ) ψ 6D Evaluate the eigenvalues λ h of the second term and solve the equation: h 2 0 = i d h hdxψ + iλ h 2ψ ψ = (const.) h λ = (const.) ( + N x ) λ TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 20

21 { N > 0 : negative tension Note that h = + N x = e φ N < 0 : positive tension Then only the negative eigenvalues λ < 0 imply the normalizable modes. h(x ) with N < 0 λ > 0 λ < 0 x Result two left-chiral normalizable modes and one right-chiral normalizable mode 2 = generation consistent with the information from deformed conifold (T-dual of this setup) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 2

22 Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions

23 Yet Another Alternative to Compactification (cf. Randall-Sundrum model) Compactify all extra directions (x,..., x 6 ) to six-torus (T 5 (S /Z 2 )) h(x ) 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 direction Then, we modify the function h to h(x ) = h 0 + N x 2πkr c, k Z TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 23

24 h(x ) x This setup has the following features: Cosmological constant vanishes Warp factor is milder (linear) than that of RS (exponential) Supersymmetry is broken We obtain 4-dim l non-susy model with E 6 gauge symmetry (under the vanishing limit of h 0 ) TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 24

25 Contents Introduction Heterotic String Theory Intersection Yet Another Model Summary and Discussions

26 Summary Studied NS5-brane and its intersection in heterotic string Obtained a simple model to yield a chiral model in four dimensions Applied it to consider a non-susy model via torus compactification Discussions Much clearer (or direct) description of massless modes Nambu-Goldstone, Higgs, gauge bosons, SUSY effective action Connecting to bottom-up model-building More understanding intersecting five-branes Comparison to type II and F-theoretical configurations via string dualities

27 Thank You