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.