1. 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 KEK
10th of November, 2009

2. 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?

3. 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

4. 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 Higgs .
Actually, this GUT model implies the E-TYPE gauge
symmetry behind.

5. 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?

6. 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

7. 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

8. 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

9. 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

10. 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!

11. (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 …

12. 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

13. 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 …

14. 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.

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

16. 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.

17. 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

18. 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!

19. 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

20. 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

21. 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

22. 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…

23. 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

24. 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

25. 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

26. 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?

27. 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 ← 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.

28. 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 → An: y2=x2+zn+1+s(u1, u2)zn
We can find the vev of Higgs field
from the singularity deformation.

29. 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!

30. 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.

31. 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 …

32. 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

33. 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!

34. 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

35. 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.

36. 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.

37. 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!

38. 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.

39. 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