Constructing the Standard Model Group in F-theory 

Kang-Sin ChoiEwha Womans UniversityTheory Seminar, KIAS, Jan. 6, 2014.

Based on arXiv:1309.7297, 1203.3812

Outline

• Introduction to F-theory– 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)

MotivationReview on Grand Unified Theory

The Standard Model

• 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

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 10 10 5 + 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.

Compactification

Large symmetry predicted by string theory10D, N=8 SUSY, E8×E8 SYM

Small observed symmetry of SM4D, N=1 SUSY, SU(3)×SU(2)×U(1) with observed field contents.

Symmetry breakingassociated withgeometric symmetry ofinternal space

F-theorycompactified on a torus gives type IIB string theory

IIB theory

• SUGRA field contents– 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 = 1generated by

F-theory

• 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

[Vafa]

𝜏 + 1

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

+ –

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 T2D

Base B6D

M3,1

Discriminant locus

• Elliptic fiber y2 = x3 + f x + g.• f and g will depend on the base coordinate.• Relation to

• Discriminant D = 4 f 3 + 27 g2

• Torus singular at D = 0– Codimension one complex surface: Sevenbrane– gs → 0: fundamental string light– gs → ∞: D-brane light

Setup summary

Fiber T2D

Base B6D

Discriminant locus4D

M3,1

IIB sevenbrane ispurely geometric object:The discriminant locusD = 0.

GeometrySingularity describes gauge theory

Strings from M2 branes

• Between 7-branes, we have spheres CP1.

• An M2 brane wrapped on it gives string in IIB limit.

Basedir.

Torus dir.T-dual to M direction

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(3)U(2)xU(1)U(1)xU(1)xU(1)

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]

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.

Model construction

• SU(2):– 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]

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]

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]…

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

My result

Blowing-up analysis of the SM singularity.– Intersection structure.

1. Proof on the SM singularity.– Not present in the table.

2. Global existence of U(1) group, using the factorization [Mayrhofer, Palti, Weigand] [Esole, Yau] method. Ex. Hypercharge and U(1)X etc.

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

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

• Properties1. The SU(3) gauge theory is localized

at w = 0: hyperspaces in B.

I3 SU(3) 0 1 1 2 3 3

I5 SU(5) 0 1 2 3 5 5

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

• Properties1. The SU(3) gauge theory is localized at w = 0 and

the SU(2) at w’ = w + a1d5 = 0: hyperspaces in B.

Construction of 4D model

2. a1 = 0 becomes SU(5), corresponding to unhiggsing by X.

3. 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

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=0forming SU(3) and SU(2) Dynkin diagram.

– e0 = 0 and w’=0 are already present butplay similar roles.

– Although Ei’s are not independent, we have disconnected SU(3)xSU(2).

E1

E0

E2 E

Resolution

• The resolved SU(3)×SU(2)×U(1) singularities:

– By – 3 resolutions plus 2 similar divisors

Global realization of U(1)

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 theoryalong 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]

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 butbut not globally.

Twogloballyconnected

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 newresolved CP1 now we call S. [Mayrhofer, Palti, Weigand] [Esole, Yau]

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

Conclusions

F-theory provides us:• 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.