F-theory and its Applications to Phenomenology Hirotaka Hayashi (U.Tokyo) Based on the work with Teruhiko Kawano (U.Tokyo), Radu, Tatar (Liverpool U.), Yukinobu Toda (IPMU), Yoichi Tsuchiya (U. Toyo), Taizan Watari (IPMU), Masahito Yamazaki (U. Tokyo) String Advanced Lectures @ KEK 10th of November, 2009

The Mystery of Yukawa Couplings ◇ The observation of Yukawa couplings has showed us very peculiar characteristics. ▷ Hierarchical Structure Mass eigenvalues have hierarchical structure among generations Ex. up-quark mass 〜 3 MeV, top-quark mass 〜 170 GeV ▷ Pairing Structure (Generation Structure) CKM matrix has three almost one entries and others are almost zero. More than that, CKM matrix is almost diagonal. Pairing Structure 〜 Generation Structure 〜 ▷ Anarchy PMNS matrix has no particular structure. Can we understand these structures from String Theory?

Bottom-up Guidelines Which vacua in String Theory should we study? ◇ The most promising solution to the quadratic divergence of Higgs mass is low energy N=1 supersymmetry. MSSM is a good candidate for beyond standard model. ▷ MSSM has in addition good features. □ It naturally includes the candidate of dark matter □ Three gauge couplings unify (i.e. GUT) at the energy scale 〜 2×1016 GeV Bottom-up Guidelines: N=1 Supersymmetry & Grand Unification Theory at High Energy

SU(5) & SO(10) GUT ◇ Matter Fields of SU(5) GUT _ 5 rep = 10 rep = Up-type Yukawa Coupling Down-type Yukawa Coupling ◇ SO(10) GUT _ 10 repr. matter, and 5 repr. matter and SU(5) singlet are put into 16 representation of SO(10). Yukawa coupling is 16-16-10Higgs . Actually, this GUT model implies the E-TYPE gauge symmetry behind.

Bottom-up Guidelines Find a vacua in STRING THEORY which has 4d N=1 supersymmetry & Grand Unification (especially E-type gauge symmetry) Does it realistic? What is the flavor structure of Yukawa couplings on that vacua?

Plan of Talk Bottom-up Guidelines Towards N=1 SUSY Vacua of 4d in String Theory What is F-Theory ? Yukawa Couplings in F-Theory Flavor Structure of Yukawa Couplings in F-Theory Summary

Heuristic Searching for N=1 SUSY GUT Vacua of 4d in String Theory Type I SO(32) Heterotic SO(32) Type IIA Heterotic E8×E8 Type IIB Most promising for GUT

Unbroken N=1 SUSY of 4d ◇ Candidate Vacuum Configurations Candelas, Horowitz, Strominger, Witten ‘85 ⇒ Compactifications of 10d SO(32)/E8×E8 supergravity/superstring Requirements: Compactifications on M4 × Z (M4 is maximally symmetric) Unbroken N=1 SUSY in 4d Realistic Gauge Group and Fermion Spectrum Bosonic fields: GMN, BMN, φ, FMN ☆ Backgrounds GMN : M4 → Minkowski, Z → Calabi-Yau 3-fold BMN : H = dB = 0 φ : φ = constant

Stability & Standard Embedding ☆ Gauge Fields Backgrounds → Holomorphic, Stable Vector Bundle V Uhlenbeck, Yau ‘85 ※ Stability For any subbundle F ⊂ V on Z, V satisfies ○ Anomaly Cancellation Condition ONE solution → Standard Embedding ↑ Spin connection

Gauge Symmetry Breaking ▷ Standard Embedding Spin connection of SU(3) holonomy manifold is a SU(3) gauge field. ↑Good for GUT ◇ Further Gauge Symmetry Breaking ⇒ Difficult! ONE way → Take quotient by freely acting group G; Z/G Candelas, Horowitz, Strominger, Witten ‘85 Witten ‘85 We can introduce non-trivial configurations of gauge fields, which has zero field strength. ○ A Bonus! For Z: quintic & G: Z5×Z5 We have FOUR generations!

(0, 2) Compactifications & Alternative Ways ◇ Compatifications w/o standard embedding → (0 , 2) cpts Hull, Witten ‘85 Distler ‘87, Distler, Greene ‘88 ▶ Difficult to obtain such vector bundle ▶ Take care of worldsheet instanton effects not to break supersymmetry There are also another ways… Dixon, Harvey, Vafa, Witten ‘85, 86 Ibanez, Kim, Nilles, Quevedo ‘87 Katsuki, Kawamura, Kobayashi, Ohtsubo ‘88 … ▷ Compactifications on Orbifolds Imposing twist boundary conditions on strings ⇒ Internal space R6/S S: space group ※ This is actually a special limit o CY compactifications ▷ Nongeometric Compactifications Kawai, Lewellen, Tye ‘86 Antoniadis, Bachas, Kounnas ‘87 Antoniadis, Ellis, Hagelin, Nanopoulos ‘89 … □ Free Fermionic Formulations □ Gepner-Kazama-Suzuki Models Gepner ‘87, 88, Lutken, Ross ‘88 Kazama, Suzuki ‘89, Font, Ibanez, Quevedo ‘89 …

String Duality Type I SO(32) Heterotic Type IIA SO(32) M-theory String Duality has broadened our perspectives and enables us to explore much more vacua! Type I SO(32) Witten ‘95 Heterotic SO(32) Type IIA M-theory Type IIB Heterotic E8×E8

Non-perturbative Heterotic Vacua ☆ Bianchi identity with 5-brane source terms Duff, Minasian, Witten ‘96 ⇒ No need for standard embedding! ☆ Furthermore, stable vector bundle is explicitly constructed on elliptically-fibered Calabi-Yau 3-fold. Donagi ‘92 Friedman, Morgan, Witten ‘97 Bershadsky, Johansen, Pantev, Sadov ‘97 Ex. V1=SU(5), V2=0 ↑Georgi-Glashow SU(5) GUT ◇ Many Models have been constructed. Donagi, Lukas, Ovrut, Waldram ‘99, Donagi, Ovrut, Pantev, Waldram ‘99, ‘00 Braun, Ovrut, Pantev, Reinebacher, ‘04, Braun, He, Ovrut, Pantev ‘05, Bouchard, Donagi ‘05 …

Problems of Heterotic Vacua ▷ Observational Problem □ Small hierarchy between GUT scale and Planck scale ＊ 4d effective lagrangian Witten ‘96 coupling constants become… ⇒ We can solve this problem in the strong coupling region. ▷ Moduli Fixing Problem □ There is no known mechanism to fix all the moduli, especially vector bundle moduli, in Heterotic String Theory. ⇒ We can fix all the moduli in type IIA/B String Theory.

What is the Good Vacua? Type I SO(32) Heterotic Type IIA SO(32) M-theory F-theory ! Type IIB Heterotic E8×E8

F theory 〜 Type IIB string theory What is F Theory? (Vafa ‘96) F theory 〜 Type IIB string theory We compactify Type IIB string theory on a background where complex coupling constant is NOT constant, but is holomorphic. We get N=1 4d effective theory. τ has SL(2, Z) symmetry in Type IIB string theory, so, we can think of τ as compex structure of torus.

T2-Fibration ← 2 dim CY4 F Not CY3 IIB ← 6 dim ★Therefore, we can think of the background as T2 fibration over 6 dimensional manifold whose torus shape is different from place to place. ☆ A sketch of an internal manifold ← 2 dim Insert the figure of torus fibration of a manifold. A torus is usual one and another is singular. CY4 F Not CY3 IIB ← 6 dim

What is happening when fiber is singular? Singular Fiber What is happening when fiber is singular? At a singular point zi, complex structure behaves as . So, when circling around z = zi, τ undergoes monodromy. C0 is a magnetic charge of D7-brane, So, we deduce that there is a D7-brane at z = zi!

Singular Pattern D7 brane (p, q) 7-brane Insert the figure of torus fibration of a manifold. A torus is usual, other one is singular, and the other torus is singular with pa+qb cycle degenerated. We also put the table which indicate where is 7-brane. D7 brane (p, q) 7-brane

Gauge Group and Chiral Matter Using (p, q) 7-brane, we can get an ADE gauge group. F theory 〜 7-brane with ADE gauge group ▷ Gravity can be decoupled from the gauge theory. ▷ We have E-type gauge symmetry. ⇒ ◇ Chiral matter is localized along the intersection of 7-branes (called as a matter curve) and it can be identified as a holomorphic section of the line bundle using Heterotic-F theory duality. Donagi, Wijnholt ‘0802 H.H., Tatar, Toda, Watari, Yamazaki ‘0805 Matter curve 7-brane

Intrinsic Formulation of F-Theory Beasley, Heckman, Vafa 0802, 0806 ◇ They constructed low energy effective theory on 7-branes (8d) and also effective defect theory (6d) coupled o bulk YM theories. Remarkably, this formulation partially reproduces the previous results obtained using Heterotic- F theory duality. ☆ Fields contents of low energy effective theory on 7-branes SUSY ↔ ↑Partial twisting to preserve N=1 SUSY in 4d ☆ e.o.m ☆Superpotential

F-theory and Yukawa couplings ▷ Now chiral matters in F-theory are identified. What about Yukawa couplings??? Yukawa couplings seem to be generated from the intersection of matter curves… i.e. a point in an internal manifold ☆ A sketch of matter curves ↓ Yukawa?? Yukawa?? → Thus, we need microscopic formulations to compute Yukawa couplings. However, it was difficult to analyze them because there is no worldsheet formulations in F-theory…

Gauge Symmetry and Singularities ◇ Remember that 7-brane is localized where torus fiber degenerates. elliptically-fibered surface 2 dim → ← 4 dim 6 dim → ← 4 dim Ex. SU(5) 5 D7-branes ↔ A4 singularity

ADE Singularities ☆ Surface singularities are completely classified as ADE type. When we blow up ADE singularities, we have ADE Dynkin diagrams respectively. (An → SU(n+1), Dn → SO(2n), E6,7,8) Ex. An Singularity (SU(n+1)) f(x,y,z)=y2-x2+zn+1=0 (x, y, z ∈ C) ・ Singularity is defined where ▷ Blow ups of An singularity

Enhancement of Singularity ADE Gauge Symmetry on 7-brane 〜 ADE Singularity Codimension-1 singularity of 6 dim base manifold ▷ Chiral matter → localized along the intersection of 7-branes A5 singularity! (codim-2 singularity) (Morrison Vafa ‘96) (Katz Vafa ‘97) ← A4 singularity

Up-type Yukawa Singularity ◇ The information of gauge symmetry is encoded into singularity of geometry in F-theory. Tatar, Watari ‘06 Donagi, Wijnholt 0802 Beasley et al 0802 H.H. et al 0805 Ex. SU(5) GUT _ D5 sing. (10+10 rep.) ↓ 　　_ ← A5 sing. (5+5 rep.) ↑ E6 sing. (10•10•5 Yukawa??) codimension-3 sing. → ← A4 sing. (SU(5) gauge symmetry) How can we capture this information?

Geometric Singularity Deformation ☆ Geometrically, singularity deformation is captured by the deformation of defining equation. Ex. An+1 → An (SU(n+2) → SU(n+1)) An+1: y2=x2+zn+2 ← An: y2=x2+zn+2+s(u, v)zn+1 ↑ Singularity is enhanced where s(u,v)=0 Non-zero coefficient represents the deformation of singularity ↑Matter curve! ☆ On the other hand, singularity deformation means gauge symmetry breaking. Therefore we can capture this feature by Higgs mechanism too.

Field Theoretic Singularity Deformation ◇ Higgs field is a transverse fluctuation of 7-branes Ex. An+1 → An (SU(n+2) → SU(n+1)) SU(n+2) gauge symmetry → SU(n+1) gauge symmetry ＝ cf. An+1: y2=x2+zn+1 → An: y2=x2+zn+1+s(u1, u2)zn We can find the vev of Higgs field from the singularity deformation.

Recipe for Yukawa couplings in F-theory (H.H. et al 0901) ◇ Information of Higgs vev can be put into the low energy effective theory constructed by Beasley, Heckman, Vafa. 1. Singularity deformation for the desired Yukawa couplings 2. Single out the background configurations of Higgs field 3. Solve the e.o.m of wavefunction on the background 4. Insert the solutions of wavefunction into superpotential and integrate over the internal 4 dimensional manifold → Then, we can obtain 4d Yukawa couplings!

Up-type Yukawa Couplings (H.H. et al 0901) ▷ We consider E6 → A4 singularity deformation E6 singularity (10•10•5 Yukawa??) ( ) D5 singularity (SO(10)) A5 singularity (SU(6)) A4 singularity (SU(5) gauge symmetry) □ E6 → SU(5)GUT × 〈U(2)〉 ← Higgs vev ☆Up-type Yukawa Couplings in generic background ↑ Insert the background and the solution of e.o.m →This matrix is made of two vectors. Thus it is RANK ONE.

Topological Invariants of Matter Curves H.H. Kawano, Tsuchiya, Watari 0910 ◇ Consideration so far is only valid when #E6 = #D6 =1 When we have multiple number of E6 and D6 points, the rank of Yukawa matrices become min(#E6, Ngen) or min (#D6, Ngen). Actually, we can prove there are always EVEN number of E6 points! ▼ Celebrated rank 1 Yukawa structure is ruined! ☆ Typical numbers Note: This situation can be evaded by considering factorization of matter curves. Marsano, Saulina, Schafer-Nameki 0904 …

Holomorphic Frame vs Unitary Frame ☆ Background Gauge Field → we always take holomorphic frame wavefunctions are just holomorphic functions ↑ “complexified” gauge transformation Takes values at GC ↓ unitary frame □ Superpotential ← invariant under complexified gauge transformation □ Kahler potential ← NOT invariant under complexified gauge transformation To obtain “Physical” Yukawa couplings, we have to take into account D-term contributions in holomorphic frame or we consider unitary frame from the first

Ex. Genus 1 Matter Curve ◇ Assume a flat metric on genus 1 matter curve E, we consider the background gauge field in unitary frame as ☆ N independent zero modes fi under the background fj = □ ↑ Gaussian wavefunction in some limit of cpx str! ＊This gaussian wavefunction makes contribution from a point dominant. → recover rank 1 structure!

Flavor Structure from Localized Matter ONE Assumption: Tune the cpx str of 10 repr. matter curve so that wavefunction on the curve behaves like Gaussian We consider the following example. Ai: E6 point Pi: D6 point q3 q2 q1 q3 q2 q1 uc1 uc3 uc2 _ c(10) _ c(10) Solid red line: left-handed quark doublets, qi Dotted green line: anti up-type quarks uci

Hierarchical Structure -1 □ Up-type Yukawa → the largest eignvalues on a given point A Others are exponentially suppressed Ex. Point-A3 → the largest entry is (3,3) Total Yukawa consists of the largest values from each point. But there also exponential hierarchy among them. ⇒ Exponential hierarchy for Yukawa eigenvalues.

Hierarchical Structure-2 □ Down-type/(Charged lepton) Yukawa ( ) ( ) →There are exponential hierarchy among the values of for Pi (i = 0, 1, 2, 3). ⇒ Exponential hierarchy for Yukawa eigenvalues. Furthermore, hierarchical structure of down-type/charged lepton Yukawa is milder than that of up-type Yukawa.

Paring Structure and Anarchy ◇ CKM Matrix The mismatch between the mass-eigenbasis of left handed up- type quarks and that of left-handed down-type quarks. Exponential hierarchy among the values of Yukawa implies that the mass eigenstates are the largest values at a point. → Pairing Structure! ◇ PMNS Matrix On the other hand, neither lepton doublet nor down-type Higgs does not localize along the matter curve → No strucure, i.e. Anarchy!

Summary ☆ Among the vacua of string theory, F-theory is a suitable vacua to describe the N=1 supersymmetry GUT theory of 4d. ☆ Heterotic-F theory duality enables us to identify the chiral matter in F-theory in terms of geometry and G-flux. ☆ Yukawa couplings in F-theory are generated from a codimension-3 singularity point. Up-type Yukawa coupling from a point of E6 and down-type Yukawa coupling from a point of D6 are generally rank one matrix. ☆ Flavor structure of Yukawa couplings in F-theory has the realistic texture, i.e. Hierarchical structures, Pairing Structure, Anarchy, when the complex structure of matter curve of 10-representation matter is tuned so that the wavefunctions of 10-representation matter are localized.

Other Topics ▷Proton Decay → Factorization of Higgs matter curves, Beasley, Heckman, Vafa 0806 Donagi Wijnholt 0805 Factorization of spectral surfaces Z2 symmetry Tatar, Watari 0602, Tatar, Tsuchiya, Watari 0905 Marsano, Saulina, Schafer-Nameki 0906 Tatar, Tsuchiya, Watari 0905 H.H., Kawano, Tsuchiya, Watari 0910 ▷ SUSY Breaking → D-brane instanton Heckman, Marsano, Saulina, Schafer-Nameki, Vafa 0808 Marsano, Saulina, Schafer-Nameki, 0808 × 2 Heckman, Vafa 0809… ▷ Right-handed Neutrino → Kaluza-Klein modes Bouchard, Heckman, Seo, Vafa 0904 Complex structure moduli Tatar, Tsuchiya, Watari 0905