F- 理論におけるフラックスコンパクト化 Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) 2018/1/29@Kyoto Sangyo U.
Outline Introduction Flux compactification in type IIB string Flux compactification in F-theory i) F-theory ii) Setup iii) Flux compactification Conclusion
Introduction The standard model of particle physics Gauge group:su(3) SU(2) U(1) Matter content: 3
Introduction Problem: No gravitational interaction in the standard model String theory A good candidate for the unified theory of the gauge and gravitational interactions Closed string Graviton Dp-brane Ramond-Ramond field: C p+1 4
Introduction Problem: No gravitational interaction in the standard model String theory A good candidate for the unified theory of the gauge and gravitational interactions Closed string Graviton Open string Dp-brane Ramond-Ramond field: C p+1 Gauge fields 5
(Perturbative) superstring theory requires the extra 6 dimension. 10=4+6 The geometric parameters of extra 6D dimensions 4D scalar fields (called moduli) Unless they are stabilized, it will lead to unobserved fifth forces. Stabilization of the extra dimensional space Moduli stabilization (creating a moduli potential) 6D Calabi-Yau (CY)Manifold R ij = 0
Moduli are ubiquitous in string compactifications Good candidate of inflaton Supersymmetry breaking Moduli cosmology Moduli interact with matter fields gravitationally. Such long-lived particles affects the cosmology of the early Universe (e.g., dark matter abundance, baryon asymmetry, )
Yukawa couplings In string theory as well as the higher-dimensional theory, Yukawa couplings Overlap integral of matter wave functions λ Yukawa = න ψφψ CY Yukawa couplings depend on the moduli fields. From such phenomenological points of view, it is quite important to discuss the moduli dynamics.
Two types of moduli fields (4D massless scalar fields): 1Closed string moduli i) Dilaton ( τ ) Imτ = g s 1 g s : string coupling ii) Kähler moduli ( δg ij ) Size of the internal cycles iii) Complex structure moduli (δg i j ) Shape
Two types of moduli fields (4D massless scalar fields): Dp-brane Gauge fields: A μ (μ = 0,1, p) x a Scalar field : Φ a (a = p + 1,, 9)
Two types of moduli fields (4D massless scalar fields): Dp-brane 2Open string moduli A i (i = 4,5,.., p) Φ a (a = p + 1,, 9) Gauge fields: A μ (μ = 0,1, p) x a Scalar field : Φ a (a = p + 1,, 9)
Two types of moduli fields (4D massless scalar fields): 1Closed string moduli 2Open string moduli Moduli dynamics will give significant effects to our Universe. In this talk, we consider the stabilization of both the open and closed string moduli based on F-theory ( non-perturbative description of IIB string).
Outline Introduction Flux compactification in type IIB string Flux compactification in F-theory i) F-theory ii) Setup iii) Flux compactification Conclusion
Flux compactification Flux compactification is useful to stabilize the moduli fields. Let us consider higher-dimensional Maxwell s theory on R 1,3 M, න F p F p R 1,3 M When then exists a magnetic flux F p in a cycle Σ p of M න Σ p F p = n Z It generates a potential depending on the metric of extra dimension. 14
Flux compactification in type IIB string on CY Type IIB string on R 1,3 CY, න G 3 G 3 R 1,3 CY G 3 = F 3 τh 3 : three-form The fluxes on the three-cycle of CY (Σ 3 ) generate the moduli potential, න Σ 3 F 3 In the 4D low-energy effective action, Flux-induced superpotential: න H 3 Σ 3 [Gukov-Vafa-Witten 99] W τ, z = න G 3 Ω z CY Ω(z) : hol. (3,0) form of CY z: Complex structure moduli 15
Low-energy effective action described by 4D N=1 SUGRA K = ln i Ω ഥΩ ln i τ τ 2 ln(v(t)) W τ, z = CY G 3 τ Ω z Ω(z) : hol. (3,0) form of CY V(T) : CY Volume V = e K ቌ K IJ D I WD I,J=τ,z JW + (K T തT K T K തT 3) W 2 = 0 No-scale structure ቍ D I = I + K I K I = I K in the reduced Planck unit M pl = 1 Dilaton and complex structure moduli are stabilized at D τ W = D z W = 0 16
Low-energy effective action described by 4D N=1 SUGRA K = ln i Ω ഥΩ ln i τ τ 2 ln(v(t)) W τ, z = CY G 3 τ Ω z Ω(z) : hol. (3,0) form of CY V(T) : CY Volume V = e K ቌ K IJ D I WD I,J=τ,z JW + (K T തT K T K തT 3) W 2 = 0 No-scale structure ቍ D I = I + K I K I = I K in the reduced Planck unit M pl = 1 Dilaton and complex structure moduli are stabilized at D τ W = D z W = 0 17
Dilaton and complex structure moduli are stabilized at D τ W = D z W = 0 W τ, z = න G 3 τ Ω z CY G 3 -fluxes are constrained as the imaginary self-dual fluxes: G 3 = i 6 G 3 Tadpole condition for C 4 : G 3 = F 3 τh 3 : three-form න H 3 F 3 + Q D3 = 0 CY 18
Dilaton and complex structure moduli are stabilized at D τ W = D z W = 0 W τ, z = න G 3 τ Ω z CY G 3 -fluxes are constrained as the imaginary self-dual fluxes: G 3 = i 6 G 3 Tadpole condition for C 4 : න C 4 G 3 G 3 10D G 3 = F 3 τh 3 න C4 : three-form න H 3 F 3 + Q D3 = 0 CY 19
How do we compute the flux-induced superpotential? W = න G 3 Ω CY G 3 = F 3 τh 3 : three-form Let us expand G 3 and Ω on the integral symplectic basis (α a, β a ) of H 3 (CY, Z), F 3 H 3 G 3 = M a α a N a β a τ( M a α a N a β a ) α a β b = δ a b a, b = 0,1,, h 2,1 Period vector : Ω = X a α a F a β a Π t = F a = F X a න A a Ω, න B a Ω = (X a, F a ) Quantized fluxes (A a, B a : basis of 3-cycles in CY)
How do we compute the flux-induced superpotential? W = න G 3 Ω = n F τn H Π CY G 3 = F 3 τh 3 : three-form Let us expand G 3 and Ω on the integral symplectic basis (α a, β a ) of H 3 (CY, Z), F 3 H 3 G 3 = M a α a N a β a τ( M a α a N a β a ) α a β b = δ a b a, b = 0,1,, h 2,1 Period vector : Ω = X a α a F a β a Π t = F a = F X a න A a Ω, න B a Ω = (X a, F a ) Quantized fluxes (A a, B a : basis of 3-cycles in CY)
Period integral Π t = න Ω, න Ω = (X a, F a ) A a B a can be exactly calculated by solving the Picard-Fuchs (PF) equation. For mirror quintic CY PF equation: z 5ψ 5 P ψ (h 2,1 = 1) 5 = x 5 i ψx 1 x 2 x 3 x 4 x 5 = 0 in an orbifold of CP 4 i=1 Complex structure moduli x 1, x 2, x 3, x 4, x 5 λx 1, λx 2, λx 3, λx 4, λx 5 ൭z d dz ൱ 4 5z ൭5z d ൱ dz + 1 ൭5z d dz + 2൱ ൭5z d dz + 3൱ ൭5z d dz + 4൱ Π = 0 Δ Ω = න P=0 P(ψ) Δ: 4-form in CP 4 [Candelas et al, 91]
Period integral Π t = න Ω, න Ω = (X a, F a ) A a B a can be exactly calculated by solving the Picard-Fuchs (PF) equation. For mirror quintic CY PF equation: z 5ψ 5 P ψ (h 2,1 = 1) 5 = x 5 i ψx 1 x 2 x 3 x 4 x 5 = 0 in an orbifold of CP 4 i=1 Complex structure moduli x 1, x 2, x 3, x 4, x 5 λx 1, λx 2, λx 3, λx 4, λx 5 ൭z d dz ൱ 4 5z ൭5z d ൱ dz + 1 ൭5z d dz + 2൱ ൭5z d dz + 3൱ ൭5z d dz + 4൱ Π = 0 E.g., in the large complex structure limit ψ 1 (z 1), Π 0 1 + Π 2 5 ln 2 z 2 2πi 2 + ln z Π 3 5 ln 3 z Π 6 2πi 3 + 1 2πi + [Candelas et al, 91]
4D effective potential: Previous mirror quintic CY can be engineered by 2D N=(2,2) U(1) Gauged Linear Sigma Model with 6 chiral superfields Φ 0,1,,5 Picard-Fuchs operator: W = න G 3 Ω = n F τn H Π CY K = ln Π i Σ ij Π j ln i τ τ U(1) charges (Toric charges) l = (l 0, l 1, l 2, l 3, l 4, l 5 ) = 1 D = a 0 Π li >0 ൭ l i ൱ Π a li<0 i ൭ ൱ a i 2 ln(v(t)) Σ ij : Symplectic matrix Stabilization of complex structure moduli and dilaton 5,1,1,1,1,1 l i z 1 l0πi=0 n [Witten 93] a i l i [Hosono-Klemm-Theisen-Yau, 93]
Comment on the Kähler Moduli stabilization The remaining Kähler moduli (T) can be stabilized by the nonperturbative effects. W =< W flux > +Ae at A, a : Constants KKLT scenario (< W flux > 1 ) LARGE volume scenario (< W flux > O(1) ) [Kachru-Kallosh-Linde-Trivedi 03] [Balasubramanian-Berglund-Conlon-Quevedo 05] De Sitter vacua can be realized by introducing the anti D3-branes. Radiative moduli stabilization scenario [Kobayashi-Omoto-Otsuka-Tatsuishi 18] 25
Outline Introduction Flux compactification in type IIB string Flux compactification in F-theory i) F-theory ii) Setup iii) Flux compactification Conclusion
Type IIB action in Einstein frame(with other fields set to 0): L IIB = g ൭R τ 2 ൱ 2(Im τ) 2 where τ = C 0 + ie φ, This action is invariant under SL 2, Z : ( Imτ = g s 1, g s : string coupling) τ τ + 1, τ 1/τ
Type IIB action in Einstein frame(with other fields set to 0): L IIB = g ൭R τ 2 ൱ 2(Im τ) 2 where τ = C 0 + ie φ, This action is invariant under SL 2, Z : ( Imτ = g s 1, g s : string coupling) τ τ + 1, τ 1/τ Interpret τ as complex structure of auxiliary torus T 2 (Vafa 96) T 2 = τ τ + 1 0 1
F-theory is defined in 12 D spacetime 12=4+8 u 0 u τ(u) 6D manifold τ u : dilaton g s = Im τ 1 g s : string coupling 8D CY manifold String coupling can be taken as g s = Im τ 1 > 1. F-theory = non-perturbative description of type IIB
D7-brane looks like cosmic string in ambient space (Greene, Shapere, Vafa, Yau, 89) Metric: 7 ds 2 10 = dt 2 + dx 2 i + H u, തu dud തu i=1 D7 D7-brane has magnetic charge under C 0 1 = ර u=u 0 dc 0 = C 0 ue 2πi C 0 u u 0 τ D7-brane
D7-brane looks like cosmic string in ambient space (Greene, Shapere, Vafa, Yau, 89) Metric: 7 ds 2 10 = dt 2 + dx 2 i + H u, തu dud തu i=1 D7 D7-brane has magnetic charge under C 0 1 = ර dc 0 = C 0 u=u 0 ue 2πi C 0 u C 0 = Reτ Near D7-brane : τ 1 2πi 0 D7-brane location : τ u 0 i (T 2 degenerate at u = u 0.) u 0 τ D7-brane
D7-brane looks like cosmic string in ambient space (Greene, Shapere, Vafa, Yau, 89) Metric: 7 ds 2 10 = dt 2 + dx 2 i + H u, തu dud തu i=1 D7 D7-brane has magnetic charge under C 0 1 = ර dc 0 = C 0 u=u 0 ue 2πi C 0 u C 0 = Reτ Near D7-brane : τ 1 2πi 0 D7-brane location : τ u 0 i (T 2 degenerate at u = u 0.) u 0 u u 0 τ D7-brane τ(u) 6D manifold 8D CY manifold
F-theory is defined in 12 D spacetime 12=4+8 u τ(u) 3D Kähler manifold τ u : dilaton g s = Im τ 1 g s : string coupling 17-branes exist at the singular limit of torus Elliptically fibered CY fourfold (CY4) 2String coupling > 1 ( Non-perturbative description of type IIB superstring) 3 Both open and closed string moduli are involved.
Outline Introduction Flux compactification in type IIB string Flux compactification in F-theory i) F-theory ii) Setup iii) Flux compactification Conclusion
Brane Superpotential: W brane = න Ω Γ, Γ=C For branes wrapping on the whole CY, open string partition function is given by holomorphic Chern-Simons theory [Witten 92]: W = න Ω Tr[A CY A + 2 A A A] 3 Lower dimensional branes wrapping on holomorphic submanifold C can be obtained by dimensional reduction A φ [Aganagic-Vafa 00] W brane (ψ, φ) = Γ, Γ=C Ω Γ CY C
Brane superpotential (Open mirror symmetry) Mirror Quintic CY3 (degree 5 hypersurface in CP 4 ) Γ P ψ = σ5 i=1 x i 5 5ψx 1 x 2 x 3 x 4 x 5 = 0 Let us consider holomorphic 2-cycles where the brane wraps[morrison-walcher 07] C ± : x 1 + x 2 = 0, x 3 + x 4 = 0, x 5 2 ± 5ψx 1 x 3 = 0 W = න Ω No moduli dependence at fixed C ±! Brane deformation: Γ into (generically non-holomorphic) curve surrounded by a holomorphic divisor Γ C + C Mirror Quintic CY3
Brane superpotential (Open mirror symmetry) Mirror Quintic CY3 (degree 5 hypersurface in CP 4 ) Γ P ψ = σ5 i=1 x i 5 5ψx 1 x 2 x 3 x 4 x 5 = 0 Continuous deformation of C ± : (Hol. divisor defined by a degree 4 polynomial) Q φ = x 5 4 5φx 1 x 2 x 3 x 4 = 0 C C + φ = 5ψ φ = + 5ψ Mirror Quintic CY3 Brane superpotential: W brane ψ, φ Brane deformation = නΩ(ψ, φ) = න which is related to D7-brane with magnetic flux F Γ Γ, Γ=Q φ F Ω [Grimm-Ha-Klemm-Klevers 09]
Toric charges of the previous system, l = 5,1,1,1,1,1; 0,0 l = 1,0,0,0,0,1; 1, 1 l: Quintic CY3 l : brane deformation The period integral Π i = න Ω ψ, φ Γ i can be computed by solving the corresponding Picard-Fuchs equation. Brane and geometry cannot be distinguished. The above system is a noncompact CY4. (CY3 fibered over ) [Jockers-Soroush 08] Compactification CP 1 leads to a compact CY4.
CY3+brane CY4 without brane In the toric language, the previous system corresponds to A-model : Quintic CY3 over CP 1 [Berglund-Mayr 98, Grimm-Ha-Klemm-Klevers 09, Jockers-Mayr-Walcher 09] l 1 = 4,0,1,1,1,1, 1, 1,0 l 2 = 1,1,0,0,0,0,1, 1,0 l 3 = 0, 2,0,0,0,0,0,1,1 l 1 + l 2 : Quintic CY3 l 2 : brane deformation l 3 : base CP 1 B-model : Elliptically fibered CY4 [Berglund-Mayr 98]
F-theory compactification on elliptically fibered CY4 l 1 = 4,0,1,1,1,1, 1, 1,0 l 2 = 1,1,0,0,0,0,1, 1,0 l 3 = 0, 2,0,0,0,0,0,1,1 l 1 + l 2 : Quintic CY3 l 2 : brane deformation l 3 : base CP 1 Period vector of CY4 in the large complex structure limit z: Quintic modulus S: Dilaton z 1 : Open string modulus Π i = γ i Ω : Fourfold periods γ i : Homology basis of H 4 H CY4, Z
F-theory compactification on CY4 u 0 u τ(u) τ u : dilaton 3D Kähler manifold CY3+branes Complex structure moduli of CY3 Dilaton Open string (position) moduli Elliptically fibered CY4 Complex structure moduli of CY4
Outline Introduction Flux compactification in type IIB string Flux compactification in F-theory i) F-theory ii) Setup iii) Flux compactification Conclusion
F-theory on elliptically fibered CY4 4D N=1 supergravity In 4D N=1 SUGRA Kähler potential: Superpotential: K = ln න Ω ഥΩ 2ln V CY4 = ln(π i η ij ഥΠ j ) 2ln V W = න G 4 Ω = n i η ij Π j CY4 c [Gukov-Vafa-Witten, 99] Scalar potential: V = e K ቌ I,J c K I J D I WDJW ቍ Π i = γ i Ω : Fourfold periods n i = γ i G 4 : Quantized four-form fluxes γ i : Homology basis of H 4 H CY4, Z η ij : Topological intersection matrix V : Volume of 3D Kähler base
Flux compactification in F-theory on CY4 G 3 -flux superpotential + brane superpotential in type IIB =G 4 -flux superpotential in F-theory W = න G 4 Ω CY4 [Grimm-Ha-Klemm-Klevers 09, ] Imaginary self-dual three-form fluxes in type IIB =correspond to self-dual G 4 -fluxes G 4 = G 4 [Gukov-Vafa-Witten 99] Tadpole conditions χ 24 = n D3 + 1 2 න G 4 G 4 CY4 [Becker-Becker 96] [Sethi-Vafa-Witten 96] χ: Euler number of CY4 n D3 : # of D3
F-theory compactification on elliptically fibered CY4 z: Quintic modulus S: Dilaton z 1 : Open string modulus n i : Quantized fluxes Kähler potential: NLO in g s correction Superpotential:
F-theory compactification on elliptically fibered CY4 z: Quintic modulus S: Dilaton z 1 : Open string modulus n i : Quantized fluxes Kähler potential: NLO in g s correction Superpotential: The self-dual G 4 fluxes
Vacuum structure of F-theory As a consequence of the self-dual condition to G 4 fluxes, all the moduli fields are stabilized at VEVs: D S W = D z W = D z1 W = 0 z: Quintic modulus S: Dilaton z 1 : Open string modulus n i : Quantized fluxes
Vacuum structure of F-theory Although the fluxes are constrained by the tadpole condition, χ 24 = n D3 + 1 2 න G 4 G 4 CY4 χ=1860: Euler number of CY4 n D3 : # of D3 we find the consistent F-theory vacuum, e.g., All the moduli fields can be stabilized around the LCS point of CY fourfold The masses of all the moduli fields are positive definite.
Comment on other models in F-theory on CY4 The orientifold limit of F-theory [Dasgupta-Rajesh-Sethi 99, Denef-Douglas-Florea-Grassi-Kachru 05] K3 K3 background [Berglund-Mayr 13] Elliptically fibered CY4 in the large complex structure limit [Honma-Otsuka 17]
Conclusion Mirror symmetry techniques can be applied to the F-theory compactifications. We explicitly demonstrate the moduli stabilization around the large complex structure point of the F-theory fourfold. All the complex structure moduli can be stabilized at the Minkowski minimum. Discussion Quantum corrections to the moduli potential Other CY4 Particle spectra in global F-theory models Heterotic/F-theory duality