The University of Tokyo, Hongo: October 26, 2009 Yet Another Alternative to Compactification by Heterotic Five-branes arxiv: 0905.285 [hep-th] Tetsuji KIMURA (KEK) Shun ya Mizoguchi (KEK, SOKENDAI)
Introduction
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
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
Yet Another Alternative: Intersecting 5-branes in hetertotic string theory Our Model 0 2 3 4 5 6 7 8 9 5-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
Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions
Contents Introduction Heterotic String Theory 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 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 2 +... TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 8
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
Contents Introduction Heterotic String Theory 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 (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 + ) 0 2 2 ( )( )( )( ) q = 3 good dimensions to consider four-dimensional spacetime TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION
0 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 = 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) 023 + our new idea TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 2
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 3 2 0 B @ 0 B @ + + ± ± 2h C A, (ω ±) 4 = h 2h 3 2 C A, (ω ±) 6 = h 2h 3 2 0 B @ 0 B @ 0 B @ +2 + + 2 ± ± ± ± 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
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
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
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
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
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 2 2 2 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
Spin connections (ω 3 H) m a b ω m a b are expressed as bω = 0, bω 2 = h bω 3 = h 2h 3 2 0 B @ + 3 3 2h C A, bω 4 = h 2h 3 2 0 B @ 0 B @ +2 + 2 3 3 3 3 3 3 C A C A bω 5 = h 2h 3 2 0 B @ + 3 3 C A, bω 6 = h 2h 3 2 0 B @ + 3 3 C A h = h/ x = x : step function TETSUJI KIMURA YET ANOTHER ALTERNATIVE TO COMPACTIFICATION 9
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 2 2 0 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
{ 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
Contents Introduction Heterotic String Theory Intersection Yet Another Alternative to Compactification Summary and Discussions
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
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
Contents Introduction Heterotic String Theory Intersection Yet Another Model Summary and Discussions
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
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