1. Constructing the Standard Model Group in F-theory
Based on arXiv: ,
Kang-Sin Choi
Ewha Womans University
Theory Seminar, KIAS, Jan. 6, 2014.

2. Outline Introduction to F-theory My result: A father of IIB theory
Use of torus fiber as extra dimensions
D-branes and strings are lifted to M-branes and geometry
Geometric singularity describes gauge theory
Resolving (smoothening) singularities
My result:
Construction of the Standard Model group SU(3)xSU(2)xU(1)
Resolution analysis
Globally valid U(1)

3. Motivation Review on Grand Unified Theory

4. The Standard Model 3 × q (3,2)1/6, uc (3*,1)2/3, l (1,2)1/2, …
Gauge theory
particles and interactions into numbers [Heisenberg] [Weyl]
Our Universe is based on
Particular group: force
SU(3)×SU(2)×U(1)
Particular representations: matter and Higgs
3 × q (3,2)1/6, uc (3*,1)2/3, l (1,2)1/2, …
1 × hd (1,2)1/2
Miracles
Minimal anomaly-free chiral multiplets
Charge quantization
Gauge coupling unification

5. Grand Unification explains the miracle
Hint: gauge theory based on groups and reprs.
SU(3)×SU(2)×U(1) ⊂ SU(5)
10 → q (3,2)1/6, uc (3*,1)2/3, ec (1,1)1
5* → dc (3*,1)1/3, l (1,1)-1/2
Similar for Higgses, into 5* (and 5) (triplet Higgs)
Yukawa coupling unified Yu Yd,e 10 5* 5*
Spontaneous symmetry breaking by VEV of a scalar e.g.<24>:
Chirality does not change: all extra fields are vectorlike
Quantized charges, single gauge coupling
Anomaly free theory gives anomaly free theory.

6. Compactification Small observed symmetry of SM
Large symmetry predicted by string theory
10D, N=8 SUSY, E8×E8 SYM
Symmetry breaking
associated with
geometric symmetry of
internal space
Small observed symmetry of SM
4D, N=1 SUSY, SU(3)×SU(2)×U(1) with observed field contents.

7. F-theory compactified on a torus gives type IIB string theory

8. IIB theory SUGRA field contents Action in string frame
NSNS: graviton g, dilaton φ, Kalb-Ramond two-forms B2
RR: even-forms C0, C2,…
Action in string frame
𝜏 = C0 + i eφ, B2 - 𝜏 C2,…
String SL(2,Z) symmetry ad – bc = 1 generated by

9. F-theory SL(2,Z): Symmetry of a torus with compex structure 𝜏.
[Vafa]
SL(2,Z): Symmetry of a torus with compex structure 𝜏.
Regard IIB string as ‘F-theory’ compactified on a torus.
(11+1) D →(19+1) D.
Only comp. struct. Vol T = 0.
Type IIB on non-Calabi-Yau manifold
𝜏 + 1

10. Torus Described by elliptic equation a x3 + b y3 + … = 0 in C2
Weierstrass form
y2 = x3 + f x + g.
A hypersurface in C2 including ∞
Order 2 branch points (=2 sheets) at 4 points + –

11. Compactification We further compactify IIB on a threeC base B
Compactificataion of F (12D) on
Calabi-Yau fourfold: 2+6 real dimensional
Fibration: T is not direct product
but globally well-patched
Fiber T 2D
Base B 6D
M3,1

12. Discriminant locus Elliptic fiber y2 = x3 + f x + g.
f and g will depend on the base coordinate.
Relation to
Discriminant D = 4 f g2
Torus singular at D = 0
Codimension one complex surface: Sevenbrane
gs → 0: fundamental string light
gs → ∞: D-brane light

13. Setup summary IIB sevenbrane is purely geometric object:
The discriminant locus
D = 0.
Fiber T 2D
Base B 6D
Discriminant locus 4D
M3,1

14. Geometry Singularity describes gauge theory

15. Strings from M2 branes Between 7-branes, we have spheres CP1.
An M2 brane wrapped on it gives string in IIB limit.
Base
dir.
Torus dir.
T-dual to M direction

16. Singularity Actual shape incl. base geometry.
When string shrinks, the geometry becomes singular.
Gauge symmetry enhancement. Emergence of W± bosons.
Larger gauge group: ‘sharper singularity’
U(2)xU(1)
U(3)
U(1)xU(1)xU(1)

17. Math vs physics
Mathematicians had classified the codimensionR two singularities. [Du Val] [Neron] [Kodaira] [Miranda]…
Correspondence: the intersection numbers are identical to Cartan matrix of Lie algebras. [McKay]
A, D, E algebra.
Ex. SU(3)
M/F-theory related them
M2-branes wrapping on spheres gives charged W± bosons.
B,C,F,G algebra made possible [Aspinwall, Gross]

18. Resolution Replacing a singularity with sphere(s).
CP1 = C1 ⋃ {∞}
Ex. An SU(2) singularity P = x y – z2 = 0.
At (x, y, z) = (0,0,0), grad P = (0,0,0): singular
Remove this point.
Introduce a new coordinate e for CP1 such that (x, z) = (x’ e, z’ e).
Now the original singularity (x,y,z) = (0,0,0) is replaced by e = 0 (and y = 0).
Proper transformation
P = x’ y – e z’2 = 0 is a new smooth manifold.

19. Model construction SU(2): In general Model construction:
x y = z2 or y2 = x3 + b1 z y + b2 z x + b3 z2.
In general
y2 = x3 + a1 xy + a2 x2+ a3y +a4 x +a6
Model construction:
Preparing Calabi-Yau manifold with a desired singularity.
Only simple groups are known.
We want to directly construct SU(3) × SU(2) × U(1)
[Bershadsky, Intrilligator, Kachru, Morrison, Sadov, Vafa]
[Kodaira] [Neron] [Aspinwall, Gross]

20. SUSY E8 unifies gauge bosons
GUT with E-series: E8 is the terminal group.
The SM group is E3×U(1).
10D pure SYM: gauge-matter-Higgs unification
Under E8 → SU(3) × SU(2) × S[U(5) × U(1)Y]

21. E8
Blowing down: y2 = x3 + z8
for finite gs, a codimension two 7-brane with F and D charges.
String junction with more than two ends: multi-fundamental representations and exceptional group possible [DeWolfe, Zwiebach]…

22. Why important?
The construction of SU(3)xSU(2)xU(1) is possible at Mst.
No need for complicated symmetry breaking from GUT group
Gauge coupling unification
Some symmetry breaking induces corrections to gauge coupling, around Mst, ruining the unification relation.
Nonperturbative effect in the EWSB sector.
Another Higgs in the Next-to-Minimal Supersymmetric Standard Model superpotential naturally follows
W = S Hu Hd + Sn.
Tracked as the singlet in E6 GUT

23. My result Blowing-up analysis of the SM singularity.
Intersection structure.
Proof on the SM singularity.
Not present in the table.
Global existence of U(1) group, using the factorization [Mayrhofer, Palti, Weigand] [Esole, Yau] method. Ex. Hypercharge and U(1)X etc.

24. Direct construction of SU(3)×SU(2)×U(1) just designing singularity

25. The SU(3)×SU(2)×U(1) singularity
The SU(3)×SU(2)×U(1) singularity is obtained by tuning the coeffs ai. [Choi, Koabayshi] [Choi]
y2 = x3 + a1 x y + a2 x2 + a3 y + a4 x + a6
Properties
The SU(3) gauge theory is localized at w = 0: hyperspaces in B.

26. I SU(3)
I SU(5)

27. The SU(3)×SU(2)×U(1) singularity
The SU(3)×SU(2)×U(1) singularity is obtained by tuning the coeffs ai. [Choi, Koabayshi] [Choi]
y2 = x3 + a1 x y + a2 x2 + a3 y + a4 x + a6
Properties
The SU(3) gauge theory is localized at w = 0 and the SU(2) at w’ = w + a1d5 = 0: hyperspaces in B.

28. Construction of 4D model
a1 = 0 becomes SU(5), corresponding to unhiggsing by X.
At the intersections, the correct matter fields are localized.
CodimC 1 curve localizing matter fields.
Magnetic flux along the ‘flavor brane’ -> induced flux along the curves.
Chiral 4D matter spectrum
# generations as # zero modes

29. Blown-up sphere and their intersections
After the full resolution, the SU(3)×SU(2)×U(1) model becomes completely smooth,
We have 3 resolutions e1 = 0, e2 = 0, e=0 forming SU(3) and SU(2) Dynkin diagram.
e0 = 0 and w’=0 are already present but play similar roles.
Although Ei’s are not independent, we have disconnected SU(3)xSU(2). E0 E1 E2 E

30. Resolution The resolved SU(3)×SU(2)×U(1) singularities: By
3 resolutions plus 2 similar divisors

31. Global realization of U(1)

32. Global existence of U(1)
In F-theory gauge theory is obtained as follows.
U(1) and Cartan subalgebra of nonabelian
Kaluza-Klein reduction of rank 3 antisymm tensor in M/F theory along rank 2 antisymm tensor
CMNP = ∑ AM wNP
Off-diagonal components W±
wrapped M2 brane along the shrunken CP1
We need U(1) for hypercharge and nonperturbative superpotential.
So far the description was approximate. Or used heterotic duality. [Choi, Hayashi]

33. Relation to group theory
Look at the elliptic equation at y2 = x3, or t2 = x
~ (t – t1) (t – t2) (t – t3) (t – t4) (t – t5) (t – t6)
Parameterizing the locations of the ‘flavor’ 7-branes
For the broken part: the commutant of the SM group S[U(1)xU(5)] in E8
Monodromy: for generic coefficients, locally factorized but but not globally.
Two
globally
connected

34. Factorization Adjusting the coefficient Global factorization
The first factor parameterizes U(1) component.
A new sphere
The resulting CY is singular, of a form x y = z1 z2
so we do small resolution to obtain a new resolved CP1 now we call S. [Mayrhofer, Palti, Weigand] [Esole, Yau]

35. The new resolved space S
The new CP1 space S has the intersection structure
The right group and Lorentz structure.
The off-diagonal components X,Y are massive.
Therefore, we achieved.
Global existence of U(1) generator, since obtained by the resolution.
Gauge coupling unification explicit: the conventional SU(5) GUT rel. E1 E2 S E

36. Conclusions F-theory provides us: What is new
Geometric origin of gauge theory
Exceptional group appropriate for GUT
What is new
Construction of the SM gauge group SU(3)×SU(2)×U(1).
Globally valid U(1) is found.

37. SO(10) unifies all the matter fields
A minimal simple group having an anomaly-free single matter representation.
16 = * + 1
Higgs in antoher repr. 10 = 5 + 5*
One Yukawa coupling 16•16•10 → hd l ec + hu l vc + …
Prediction: the same generations of extra neutrinos.
Field theoretic SU(5) GUT is almost ruled out but SO(10) GUT is not, with some complicated Higgs interactions.
Right-handed neutrino

38. SUSY E6 unify Higgs Supersymmetry: hd(2,1) belongs to hypermultiplet.
All the matter-Higgs multiplet unifies to one single repr.
27 = 16matter + 10Higgs + 1
Gauge invariant interaction 27•27•27
gives all the above Yukawa couplings and
S Hd Hu
<S> ~ µ in MSSM [Langacker, Wang] [King et al]
Singlet extension of MSSM: Dynamics of mainly soft SUSY breaking terms explains weak scale <S>
cf. S3 in NMSSM forbidden.