F-theory compactiﬁcations and model buildingstatistics.roma2.infn.it/~stringhe/doc/weigand.pdf ·...

F-theory compactifications andmodel building

Timo Weigand

ITP, University of Heidelberg

Tor Vergata Roma, November 2010 – p.1

3 good reasons for F-theoryI.) Two traditional foundations of string model building:

1) heterotic string:

orbifolds, free fermionic, het. on CY with vector bundles

• exceptional gauge groups X

• gravity and gauge fields all in bulk

2) perturbative Type II orientifolds/brane worlds:

Type IIA with D6-branes

Type IIB with D3/D7-branes

• perturbative gauge groups: U(N), Sp(N), SO(N) , but no E-groups

• localised gauge degrees of freedom X

F-theory combines both pros:

• model building based on exceptional groups

• includes ”local features”


3 good reasons for F-theoryII.) Promising moduli stabilisation scenarios similar to Type IIB/M-Theory

well-controlled because fluxes allow for conformal Calabi-Yau

↔ power of holomorphy

III.) F-theory is fascinating beyond phenomenology:

truly non-perturbative and backreacted compactification framework

in some sense: THE way to think about 7-branes

Dynamics accessible from several different viewpoints:

• as strongly coupled Type IIB with 7-branes

• as dual to M-theory

• as dual to heterotic strings


OverviewLecture 1: Introduction to F-theory

Lecture 2: Global Tate models and local Spectral covers for F-theory

Lecture 3: Aspects of F-theory GUT phenomenology


Overview

Part I: Introduction to F-theory

• F-theory as Type IIB with backreaction

• Elliptic fibrations and their M-theory origin

• 7-branes and non-abelian singularities


Reminder: Type IIB orientifoldsCompactify 10 D Type IIB theory on Calabi-Yau 3-fold X

quotient by Ω(−1)FLσ σ: Z2 hol. involution of X

⇒ Fixpoint set: O7-planes and O3-planes

charge cancellation requires 7-branes on Γa + image on Γa′

∑

a

Na(Γa + Γa′) = 8ΠO7

further constraints:

• induced

D5/D3-tadpole

cancellation

• D-term SUSY


Probe approximationD-branes and O-planes treated in probe approximation

• neglect backreaction on metric and on sourced RR fields

• only ensure integrability conditions = tadpole conditions

For generic Dp-brane p < 7 backreaction is asymptotically negligible

• Dp brane is pointlike source in n = 9 − p normal directions

• Heuristically: need to solve Poincare equation in n dimensions

n > 2: ∆Φ(r) = δ(r) → Φ(r) = 1rn−2

more precisiely: BPS solution

ds2 = H−1/2p ηµνdxµdxν + H

1/2p dxidxi

e2φ = e2φ0H3−p2

p Cp+1 =H−1

p − 1

eφ0dx0 ∧ . . . ∧ dxp

Hp = 1 +“ rp

r

”(7−p), r

(7−p)p = #eφ0N

Note: Hp → 1 as r → ∞ XTor Vergata Roma, November 2010 – p.7

Probe approximationIn general we wish to go beyond this probe approximation:

• While above is fine for Dp-branes, O-planes have negative tension ⇒SUGRA is ill-defined unless all tadpoles are cancelled locally

• we need control of the in general varying dilaton as this gives the

string coupling

• for the interesting case of D7-branes in Type IIB orientifolds solution

does not asymptote to flat space in above sense (see later)!

F-theory = SUSY Type IIB/Ωσ(−1)FL with backreaction of

D7-branes and O7-planes on the geometry and on the varying

dilaton taken into account


Type IIB and SL(2, Z)Consider the IIB action in Einstein frame:

SIIB =2π

ℓ8s

(∫

d10x√−gR − 1

2

∂τ∂τ

(Imτ)2+

1

ImτG3 ∧ ∗G3 +

1

2F5 ∧ ∗F5 +

+C4 ∧ H3 ∧ F3

)

,

ℓs = 2π√

α′, τ = C0 + ie−Φ, G3 = F3 − τH3, H3 = dB2,

F5 = F5 −1

2C2 ∧ H3 +

1

2B2 ∧ F3 Fp = dCp−1

Classical action is invariant under SL(2, R):

τ → aτ + b

cτ + d,

(

C2

B2

)

→(

aC2 + bB2

cC2 + dB2

)

= M

(

C2

B2

)

, detM = 1

non-perturbatively SL(2, R) → SL(2, Z) due to D(-1)-instanton

M =

(

1 b

0 1

)

: C0 → C0 + b ⇒ e2πiR

C0 invariant for b ∈ Z


Type IIB and SL(2, Z)

SL(2, Z) : T =

(

1 1

0 1

)

: τ → τ + 1 S =

(

0 1

−1 0

)

: τ → −1

τ

same symmetry group acts on complex structure of a torus T 2

τ takes values in fundamental domain

T 2: lattice in C:

w ≃ w + 1, w ≃ w + τ

τ ∈ C ℜτ > 0: complex structure1

τ τ+1

a

b

S-duality is strong-weak duality: maps F1-string to D1-string

F1: (1,0) string, D1: (0,1) string S

(

1

0

)

=

(

0

1

)

(p,q) string: BPS-bound state of p F-strings and q D-strings

stable for p,q relatively primeTor Vergata Roma, November 2010 – p.10

7-brane backreactionD7 action: S = SDBI + SCS (string frame)

SDBI = −µ7

∫

e−Φ√

−det(G + B + 2πα′F ), SCS = µ7

∫

C8 + . . .

µ7 =2π

lp+1s

|p=7 =2π

l8s

D7 brane is electric source for C8 ↔ magnetic source for C0 = Reτ

Task: find the backreacted solution for τ

D7 brane is pointlike source in 2 normal directions z = x + iy

• SUSY requires: τ = τ(z) → F1(z) = dC0(z)

• e.o.m. for F9 = dC8: d ∗ F9 = δ(2)(z − zi)

• duality ∗F9 = F1

• Gauss law: 1 =∫

Cd ∗ F9 =

∮

S1 F1 =∮

S1 dC0

IP1 IIB(z)

7-brane

z0

pic from: Lerche, 9910207

Solution for τ : τ(z) = τ0 + 12πi ln(z − zi) + . . . e−φ = 1

2π ln| z−zi

λ |Interpretation: non-trivial profile for τ , but still have unfixed offset


7-brane backreactionThis is a severe backreaction on geometry:

1) deficit angle ↔ ”long-range”effect

2) monodromies

ad 1) D7-brane looks like ”cosmic string” in ambient space

metric ansatz: ds2 = −dt2 +∑7

i=1 dx2i + H7(z, z)dzdz

• pointlike source in ambient space

⇒ Expect 2 dim. Poisson equation for H7(z, z)

• Einstein equation: ∂∂logH7 = ∂τ∂ττ22

Greene,Shapere,Vafa,Yau 1989

• at long distances away from brane, space has a deficit anglesee also Bergshoeff et al. ’06, A. Braun et al. ’08

=⇒ long-range effect that does not asymptote away

• for suitable τ0 still exists regime near brane where metric can be

treated as in probe picture

but for too strong coupling, this regime might become substringy!Tor Vergata Roma, November 2010 – p.12

7-brane backreactionad 2) near D7-brane: τ(z) ≃ 1

2πi ln(z − z0) has branch cut

as we encircle z0: τ → τ + 1 ”monodromy”

consistent because Type IIB theory is SL(2, Z) invariant

More drastic monodromy for more general (p, q) branes

• D7-brane: electric charge under C8 ↔ magnetic charge under τ

• dyonic (p,q) branes = branes on which (p, q) strings can end

• SL(2, Z) transforms various (p,q) branes into one another

We had: monodromy of (1,0) brane: τ → τ + 1 ↔ T1,0

for (p, q) brane : τ → aτ + b

cτ + dT =

(

1 − pq p2

−q2 1 + pq

)

• In general, different (p, q) type branes cannot be treated perturbatively


From branes to F-theoryF-theory geometrises the varying axio-dilaton field τ : Vafa 1996

• Interpret τ as complex structure of auxiliary torus T 2

• Kahler structure has no physical meaning

τ varies ↔ shape of T 2 varies

=⇒ fibration of T 2 → M10

pic adapted from: Denef, 0803.1194

pic adapted from: Lerche 9910207

locally near position of D-brane

τ = 12πi ln(z − z0) + . . . → i∞

↔ T 2 fiber degenerates as cycle a → 0


F from MPrecise definition of F-theory via duality with M-theory

Idea: Compactify 11-dim SUGRA on M9 × T 2

Take limit of vanishing volume of T 2 = S1A × S1

B as follows:

• S1A: M-theory circle of radius R10

R10 → 0 ↔ weakly coupled IIA on M9 × S1B

• T-duality along S1B of radius R9 → IIB on M9 × S1

B

R9 → 0 ↔ decompactification of S1B ⇒ IIB on M9 × R

Roughly: gs = R10

R9is Im(τ) for original T 2

F-theory limit is vol(T 2) → 0 while ℓs finite

Newton’s law of string theory:

F = M |A(T 2)→0

Indeed: volume of T 2 has no physical meaning in IIB


Elliptic fibrationsCase of phenomenological interest:

Type IIB orientifold on M10 = R1,3 × X3/σ with R1,3 filling 7-branes

• physics encoded in geometry of Y4 : T2 → B3 B3 = X3/σ

• N = 1 SUSY on M-theory side requires Y4 to be Calabi-Yau

• jargon: F-theory on elliptic fourfold Y4 = effective 4D theory obtained by

compactification of Type IIB strings with D7-branes on X3/σ

IIB language:

7-branes wrap 4-cycle Γa ∈ X3/σ

F-theory language:

Γa = locus of fiber degenerationpic adapted from: Denef, 0803.1194

Magic of F-theory:

Xall information of present D7-branes encoded in geometry of this fibration

Xbackreaction fully taken into account Tor Vergata Roma, November 2010 – p.16

Elliptic fibrationsUnderstand possible degenerations of fiber via algebraic geometry :

• elliptic curve admits various representations as hypersurfaces/complete

intersections in projective spaces

• most common: T 2 = P2,3,1[6]: (x, y, z) ≃ (λ2x, λ3y, λz) (∗)P ≡ y2 − x3 − fxz4 − gz6 = 0 f, g ∈ C with (∗) set z = 1

• This is Calabi-Yau and thus T 2 X

• T 2 singular when P = 0 and dP = 0, i.e.

1) y = 0

2) 0 = x3 − fx − g = (x − a1)(x − a2)(x − a3)

3) 0 = (x − a1)(x − a2) + (x − a2)(x − a3) + (x − a1)(x − a3)

⇒ only possible if two or more ai coincide


Elliptic fibrationsResults from complex geometry:

• Coincidence of roots if discriminant

∆ = 27g2 + 4f3 = 0

• complex structure τ encoded in f, g via

j(τ) =4(24f)3

∆

j(τ): SL(2, Z) invariant Jacobi function

j(τ) = e−2πiτ + 744 + O(e2πiτ )


Elliptic fibrationsNow fiber T 2 over the base B3 with coordinates ui: f, g → f(ui), g(ui)

→ Weierstrass model y2 = x3 + f(ui)xz4 + g(ui)z6

(more details on properties of B and g, f later)

position of branes on B3: ∆(ui) = 0 → holomorphic divisor X

• simple zero

only T 2 degenerates, but fourfold Y4 remains smooth

single 7-brane, possibly with U(1) gauge group

• multiple zero → genuine singularity of fourfold Y4

several coincident 7-branes with non-abelian gauge groups


ADE gauge groupsadmissable singularities of Y analysed by Kodaira ⇒ A-D-E classification

these are the singularities of Y whose resolution in fiber does not destroy

the Calabi-Yau property of Y

ord(∆) ord(f) ord(g) sing.n 0 0 An−1 ≃ SU(n)

n + 6 2 ≥ 3 Dn+4 ≃ SO(2n + 8)8 ≥ 3 4 E6

9 3 ≥ 5 E7

10 ≥ 4 5 E8

Physical interpretation:

gauge group of brane given by corresponding A-D-E group

Bershadsky et al. 1996

Note at this stage:

on general elliptic fibration mutually non-local monodromies are present

stacks of coincident (p,q) branes of different type → exceptional gauge

symmetriesTor Vergata Roma, November 2010 – p.20

ADE groups in F-theoryExample: SU(N) from N coincident branes of same (p,q) type

as N mutually local branes approach each other the fiber develops an

AN−1 singularity obtained by shrinking N P1s in fiber

YG: singular 4-fold T 2 → B3 with ADE group G along divisor S ⊂ B3

singularities best studied by resolution YG → Y G within M-theory

• paste in tree of P1s fibered over S ΓGi i = 1, . . . , rk(G)

singular YG ↔ zero size limit of ΓGi

• resolution divisors DGi ⇐⇒

fibration ΓGi → S

• Group theory of G

⇔ extended Dynkin diagram


ADE groups in F-theory2 sources of gauge bosons along S from M-theory reduction

• off-diagonal elements in ad(G):

M2-branes along chains of P1 ΓGi ∪ . . . ∪ ΓG

j , i ≤ j

=⇒ massless only in singular limit

• Cartan U(1)rk(G) generators:

3-form C3 expanded in ωGi = [DG

i ] ∈ H2(Y G, Z)

C3 =

rk(G)∑

i=1

Ai ∧ ωGi + . . . Ai ↔ gauge field along S

Gauge flux of Cartan U(1): G4 =∑

i Fi ∧ ωGi , Fi ∈ H2(S)


The quest for U(1)X Non-abelian ADE gauge symmetry built in geometrically

X Cartan U(1)s easy to study in Coulomb phase in M-theory

Extra U(1) gauge symmetries not tied to non-ab. groups

harder to detect

General fact from expansion C3 =∑rk(G)

i=1 Ai ∧ ωGi :

total rank of gauge group [Morrison,Vafa I+II ’96]

nv = h1,1(Y G) − h1,1(B) − 1

nU(1) = nv − rk(G)

↔ can be computed in global models with resolution of singularity

elliptic 3-folds: [Candelas,Font ’96] [Candelas,Perevalov,Rajesh ’97]

elliptic 4-folds: [Blumenhagen,Grimm,Jurke,TW 0908.1784][Grimm,Krause,TW 0912.3524]


Gauge groups: IIB viewpointAppearance of exceptional gauge symmetries also in strongly coupled IIB

language well-known

• origin: coincident 7-branes of mutually non-local (p,q) type

• massless degrees of freedom from string junctions:

string bound-states with multiple ends

→ graphical representation of adjoint of e.g. exceptional gauge groups

Example:

A = (1, 0), B = (3,−1), C = (1,−1) [Sen ’96/97; Gaberdiel,Zwiebach’97]

SU(N) : AN , SO(2N) : ANBC, E8 : A5BC2


F-Theory and IIBConnection to IIB orientifold on CY 3-fold X by Sen limit: [Sen ’96/97]

Weierstrass:

y2 = x3 − f(ui) x + g(ui), ∆ = 27g2 + 4f3

General parametrisation: f = −3h2 + ǫη, g = −2h3 + ǫhη − ǫ2

12χ

Generically: ∆ is single, connected I1 object

IIB limit: ǫ → 0 ⇒ ∆ = −9ǫ2h2(η2 − hχ) + O(ǫ3)

O7 : h = 0, D7 : η2 − hχ = 0

F-theory ↔ non-perturbative recombination of O-plane and 7-branes!

X : double cover of base B branched over h = 0

Simplest case: X given by equation h = ξ2, orientifold ξ → −ξ

Uplift: Reversal of Sen limit

↔ Define B = X/σ and consider Weierstrass model thereof


7-brane tadpoleall ”extra”7-brane consistency conditions of probe IIB picture

automatically incorporated in geometry: Morrison, Vafa 1996

• Kodaira: first Chern class of tangent bundle of Y descends from

tangent bundle of B modulo singular divisor loci

c1(TY ) ≃ π∗(c1(TB) −∑iai

12δ(Γi)) ai = O(∆)|Γi

• Since N = 1 SUSY requires c1(TY ) = 0∑

i

aiδ(Γi) = 12 c1(TB)

B is no longer Calabi-Yau

• cf. 7-brane tadpole:∑

i Niδ(Γi) = 4δ(O7)

→ curvature of base B has absorbed orientifold charge

elliptic Calabi-Yau fourfold ↔ consistent, non-pert. 7-brane configuration

Example: 7-branes on S2: KS = O(−2) → need 24 branesTor Vergata Roma, November 2010 – p.26

Consistency: D3 tadpoleWhile D7-tadpole automatic in F-theory, D3-tadpole ensured extra:

Consider dual M-theory on R(1,2) × Y w/ M2 branes along R(1,2) × pointi

integrate e.o.m. for G4 field strength

0 =

∫

d ∗ G4 =

∫(

1

2G4 ∧ G4 − ℓ6MI8(R) + ℓ6M

∑

i

δ(M(i)2 )

)

Key:∫

YI8(R) = χ(Y )

24 χ(Y ) =∫

Yc4(TY ): Euler characteristic

χ24 = 1

2ℓ6M

∫G4 ∧ G4 + NM2

upon M/F-theory duality:

•χ24 : curvature dependent D3-charge of O7-planes and 7-branes

• NM2 → ND3

•1

2ℓ6M

∫G4 ∧ G4: bulk and brane flux induced D3-charge


Consistency: D3 tadpoleReduction of M-theory 3-form:

C3 = C3 + B2 ∧ L dx + C2 ∧ L dy + B1 ∧ L dx ∧ L dy

IIB: Cy4 = C3 ∧ dy, giy = (B1)i, B2: NSNS 2-form C2: RR 2-form

Similarly: IIB 3-form flux from: G4 = H3 ∧ L dx + F3 ∧ L dy

• IIB bulk flux: reduce G4 on 4-cycle which is product of 1-cycle in

fiber and non-trival three-cycle Σ3 on B

↔ H3/F3 along Σ3

• IIB gauge flux F2 along the 7-brane on divisor S: G4 = F2 ∧ ω

ω anti-selfdual 2-form dual to 2-cycle from fibering pinched S1

between 2 branes


Consistency: D3 tadpoleIn absence of bulk flux:

χ

24= ND3 − 1

2ℓ4s

∫

S

F2 ∧ F2

︸︷︷︸

≥0 due to F2=−∗F2

Constraint: ND3 ≥ 0 to avoid uncontrolled SUSY breaking

Important: in presence of non-abelian gauge groups, Y is singular

χ(Y ) refers to Calabi-Yau obtained after resolving these singularities

special case: no non-abelian gauge groups ⇒ only I1 degenerations

↔ Euler character: χ∗(Y ) = 12∫

Bc1(B) c2(B) + 360

∫

Bc31(B)

[Sethi,Vafa,Witten ’96]

One can give a closed expression for certain more general configurations

required for model building [Blumenhagen,Grimm,Jurke,TW ’09]


Overview

Part II:

Global Tate models and local spectral covers

• Higher co-dimension enhancements

[Beasley,Heckman,Vafa 0802.3391, 0806.0102]

• Global Tate model

• Spectral covers as a local description of fluxes

[Donagi,Wijnholt 0802.2969, 0904.1218] , [Hayashi,Tatar,Toda,Watari,Yamazaki

0805.1057] , [Hayashi,Kawano,Tatar,Watari 0901.4941]

• global validity of spectral cover?

[Blumenhagen,Grimm,Jurke,TW ’09]; [Grimm,Krause,TW ’09]


Previously on this show....F-theory on Calabi-Yau fourfold Y4 → B3

• B3 ↔ backreacted compactification space of IIB orientifolds with

7-branes: B3 = X3/σ

• T2 fiber: varying dilaton

• ADE group G along divisor S ⊂ B3


Brane intersectionsgauge symmetry on compl. codimension-1 locus = divisor Γa on B

Γa, Γb intersect along curve Cab = Γa ∩ Γb

IIB picture: bifundamental matter

F-theory: singularity enhanced on Cab

Picture: collision of zero-size P1 in fiber over Γa, Γb

Example:

Γa: AN−1 sing., Γb: I1 sing. =⇒ Cab: AN sing.

along Cab: massless states from wrapped membranes fill adjSU(N+1)

extra states are localised as massless matter on Cab Katz, Vafa 1998

group theoretic decomposition:

SU(N + 1) → SU(N) × ”U(1)”

adjN+1 → (adjN )0 + 10 + (N)1 + (N)−1


Matter at intersections• 6D hypermultiplet along Cab in ad(Gab) ⇐⇒ (b, χα), (b, χα)

• ad(Gab) → ad(Ga) ⊕ ad(Gb) ⊕⊕

i(Ria, Ri

b)

• Matter counted by cohomology groups of 6D gauge bundle

(bi, χiα) : H0(Cab, K

12

C ⊗ VRia⊗ VRi

b),

(bi, χi

α) : H1(Cab, K12

C ⊗ VRia⊗ VRi

b)

Chiral spectrum requires gauge flux Fa, Fb

(depending on embedding this will further decompose various reps.)

χ(Ria ⊗ Ri

b) =∫

Cab

(

c1(VRia) + c1(VRi

b))


Yukawas

at intersection of two or three matter

curves: further rank enhancement

pic from: Beasley,Heckman,Vafa 0806.0102

e.g. Γ5 : SU(5), Γa : U(1)a, Γb : U(1)b

51a,0 + c.c., 50,1b+ c.c., 11a,−1b

+ c.c.

all states incorporated at point of SU(7) enhancement

there: Yukawa couplings 〈51a,0 50,−1b1−1a,1b

〉

• Yukawas descend from cubic Yang-Mills interaction for enhanced

gauge group upon decomposition of adjoint

• Holomorphic piece given by overlap of wave functions at intersection

point


Summary so farGauge dynamics described by singular fibers in ell. fibration Y4 → B3:

• ADE gauge group G along

4-cycle S ⊂ B3

• intersection Sa ∩ Sb along 2-cycle Cab:

extra massless matter

enhancement to Gab:

ad(Gab) → ad(Ga) + ad(Gb)

+∑

i

(Ui, Ri)

• intersection of three matter curves:

Yukawa couplings


Tate modelRecall: elliptic fibration Y4 : T 2 → B3 given by Weierstrass model

P : y2 − x3 − f(ui) x z4 − g(ui) z6 = 0 (x, y, z) ≃ (λ2x, λ3y, λz)

f, g vary over base B3 with coordinates ui

• x, y, f, g are sections of a line bundle L on the basis

non-triviality of L responsible for non-trivial fibration

• homogeneity of P : x ↔ L2, y ↔ L3, f ↔ L4, g ↔ L6

• discriminant: ∆ = 4f3 + 27g2 ↔ L12

L determined by Calabi-Yau condition for Y4:∑

i aiδ(Γi) = 12 c1(TB3)

δ(Γi) vanishing locus of ∆ with multiplicity ai∑

i aiδ(Γi) is in same class as ∆ (because it gives full degenerate locus)

=⇒ L = K−1B3

f ↔ K−4B3

g ↔ K−6B3


Tate modelMore convenient: Tate form

PW = x3 − y2 + x y z a1 + x2 z2 a2 + y z3 a3 + x z4 a4 + z6 a6 = 0

• local equivalence w/ Weierstrass form by completing square and cube

β2 = a21 + 4a2, β4 = a1a3 + 2 a4, β6 = a2

3 + 4a6

f = − 1

48(β2

2 − 24 β4), g = − 1

864(−β3

2 + 36β2b4 − 216 β6)

⇒ ∆ = − 14β2

2(β2β6 − β24) − 8β3

4 − 27β26 + 9β2β4β6

• precise characterisation of singularities and resulting gauge groups in

terms of Tate algorithm [Bershadsky et al. 9605200]

• Note every Weierstrass model is globally of this form (branch cuts)

Special class of models admits global Tate form and an ∈ H0(B, K−nB )

• Global Tate models: underlying E8 singularity structure: [Grimm,TW ’10]

Start with G = E8 over S and deform into G = E8/H


Tate algorithmsing. discr. group coefficient vanishing degrees

type deg(∆) enhancement a1 a2 a3 a4 a6

I0 0 — 0 0 0 0 0

I1 1 — 0 0 1 1 1

I2 2 SU(2) 0 0 1 1 2

I ns3 3 [unconv.] 0 0 2 2 3

I s3 3 [unconv.] 0 1 1 2 3

I ns2k

2k SP (2k) 0 0 k k 2k

I s2k

2k SU(2k) 0 1 k k 2k

I ns2k+1 2k + 1 [unconv.] 0 0 k + 1 k + 1 2k + 1

I s2k+1 2k + 1 SU(2k + 1) 0 1 k k + 1 2k + 1

II 2 — 1 1 1 1 1

III 3 SU(2) 1 1 1 1 2

IV ns 4 [unconv.] 1 1 1 2 2

IV s 4 SU(3) 1 1 1 2 3

I∗ ns0 6 G2 1 1 2 2 3

stolen from: [Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa 9605200]


Tate algorithmsing. discr. group coefficient vanishing degrees

type deg(∆) enhancement a1 a2 a3 a4 a6

I∗ ss0 6 SO(7) 1 1 2 2 4

I∗ s0 6 SO(8)∗ 1 1 2 2 4

I∗ ns1 7 SO(9) 1 1 2 3 4

I∗ s1 7 SO(10) 1 1 2 3 5

I∗ ns2 8 SO(11) 1 1 3 3 5

I∗ s2 8 SO(12)∗ 1 1 3 3 5

I∗ ns2k−3 2k + 3 SO(4k + 1) 1 1 k k + 1 2k

I∗ s2k−3 2k + 3 SO(4k + 2) 1 1 k k + 1 2k + 1

I∗ ns2k−2 2k + 4 SO(4k + 3) 1 1 k + 1 k + 1 2k + 1

I∗ s2k−2 2k + 4 SO(4k + 4)∗ 1 1 k + 1 k + 1 2k + 1

IV∗ ns 8 F4 1 2 2 3 4

IV∗ s 8 E6 1 2 2 3 5

III∗ 9 E7 1 2 3 3 5

II∗ 10 E8 1 2 3 4 5

stolen from: [Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa 9605200]


SU(5) GUT from TateEngineering of concrete models by suitable choice of ai

Example of phenomenological relevance: SU(5) GUT gauge group along

divisor S : w = 0

split base coordinates ui = (w, yi) and read off from Tate:

a1 = b5w0, a2 = b4w, a3 = b3w

2, a4 = b2w3, a6 = b0w

5,

bi = bi(w, yi) but no overall factor of w

y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x

2w + b5xy

∆ = − w5︸︷︷︸

5[S]

(b45P + wb

25(8b4P + b5R) + w2(16b2

3b24 + b5Q) + O(w3)

)

︸︷︷︸

[D1]

,

P = b23b4 − b2b3b5 + b0b

25, R = 4b0b4b5 − b

33 − b

22b5

[∆] = 5[S] + [D1]


SU(5) GUT from Tate

[∆] = 5[S] + [D1]

Generically: D1 does not factorise further → I1 locus

extra singular locus in addition to SU(5) unavoidable in global models


SU(5) GUT from TateMatter curves are read off as follows:

• matter in 10: Enhancement SU(5) → SO(10):

45 → 24 + 1 + 10 + 10

Tate: ∆ ≃ w7, (a1, a2, a3, a4, a6) = (1, 1, 2, 3, 5)

→ P10 : w = 0 ∩ b5 = 0

Note: fits with orientifold picture, where 10 localises on O-plane

• matter in 5: Enhancement SU(5) → SU(6):

35 → 24 + 1 + 5 + 5

Tate: ∆ ≃ w6, a1 ≃ w0

P5 : w = 0 ∩ P = b23b4 − b2b3b5 + b0b

25 = 0

In phenomenological SU(5) GUT models:

10 ↔ (QL, ucR, ec

R)

5m ↔ (dcR, L) 5H ↔ (Tu, Hu), 5H ↔ (Td, Hd)


SU(5) GUT from TateYukawa couplings from further enhancements at points:

• 〈1055〉 from enhancement SU(5) → SO(12):

〈1055〉 ⊂ 〈(66)3〉 of SO(12)

D6 point: w = 0 ∩ b5 = 0 ∩ b3 = 0

possible also perturbatively in IIB theory

• 〈10105〉 from enhancement SU(5) → E6:

〈10105〉 ⊂ 〈(78)3〉 of E6

E6 point: w = 0 ∩ b5 = 0 ∩ b4 = 0

=⇒ requires exceptional enhancements

• not possible perturbatively in IIB theory

genuine F-theory effect

• D-brane instantons do generate coupling in principle also in IIB

But different effect: E6 enhancement is not the F-theory version of

the IIB instanton effect!


Local picture: ALE fibrationsStructure of matter curves and Yukawas for states charged under SU(5)

determined only by local neighbourhood of S within Y4

Locally can think of B3 as total space of normal bundle NS/B → S

(more precisely: canonical bundle - see later)

• view normal coordinate w as parametrizing one copy of C

• T 2 fiber combines locally with w into local K3 fibration over S

Globally this is not the full story since B3 is in general not fibered over S!

Locally replace Y4 : T 2 → B3 by K3 → S

Relation to Tate model (e.g. for SU(5)):

y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x

2w + b5xy, bi = bi(w, yi)

restriction to neighbourhood of S by truncation bi → bi = bi|w=0

y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x

2w + b5xy


Local picture: ALE fibrations

Stable degeneration limit:

• view K3 = dP9 ∪ dP9 glued over

common T 2

• focus on one of the two dP9

• blow down: dP9 → dP8 pic stolen from Donagi/Wijnholt 0802.2969

dP8 = P2 with 8 points blown up to P1 Ei

=⇒ H2(dP8) = 〈l, Ei〉, i = 1, . . . 8 l2 = 1, Ei · Ej = −δij

Dynkin diagram of E8 ↔ l − E1 − E2 − E3, Ei − Ej

⇒ maximal singularity supported by dP8 lattice is E8

Only if model has heterotic dual is the 4-fold Y globally K3 fibered over S


Local picture: ALE fibrationsdP8 picture makes manifest that ALE fiber over S contains small P1

intersecting like the roots in E8-Dynkin diagram:

H2(ALE, Z) = 〈αE8

I 〉

• all αE8

I zero size ⇒ maximal enhancement E8 over S

• non-zero volume for subset αHi with Dynkin diagram H ⊂ E8

⇒ gauge group G = E8/H on S

To specify gauge group G:

Analyse root system of commutant H = E8/G


Local picture: ALE fibrationsField theoretic interpretation via 8D N = 1 SYM gauge theory on S

bosonic degrees of freedom in adjoint representation of E8 on S:

• gauge field A1,0

Cartan U(1) ⊂ E8 ↔ M-theory origin: C3 = A1,0I ∧ ωI (

∫

αIωJ = δIJ)

• Higgs field = complex scalar field

deformation modes of S

Cartan part ↔ M-theory origin: δΩ4,0 = Φ2,0I ∧ ωI

geometric Higgsing of E8 → G = E8/H by deformation of ALE fiber

↔ non-zero VEV to H-valued components of Φ

ALE fibrations: non-zero VEV only for Cartan U(1) part of Higgs field Φ

This (un)Higgsing can happen even in absence of VEV for gauge flux

Analogy in brane language: displacement of coincident branes


Spectral coversIdea of spectral covers:

describe local system by eigenvalues µi of Φ

Example: G = SU(5) → H = SU(5)⊥ Φ ↔ ad(SU(5)⊥)

• µi are the roots i = 1, . . . , 5 of det(s 15 − Φ) = 0

• Since Φ varies over S, so do the µi

• This is a degree 5 equation:

0 = s5 + e1s4 + e2s

3 + e3s2 + e4s + e5 =

∏

i

(s − µi) (∗)

• en(µi) = Tr(Φn): symmetric polynomial of degree n

• In particular: e1 =∑

i µi = Tr(φ) ≡ 0 for H = SU(5)

• More generally: Higgs field might exhibit poles

multiply by b0 to remove these poles:

0 = b0s5 + b1s

4 + b2s3 + b3s

2 + b4s + b5, en = bn

b0


Spectral coversGeometric interpretation of

0 = b0s5 + b1s

4 + b2s3 + b3s

2 + b4s + b5 = b0

∏

i

(s − µi) (∗)

• since Φ is valued in bundle KS → S: degree 5 equation in KS → S

• s = 0 is location of S in total space KS → S

• (∗) is a degree 5 cover of S in KS called the spectral cover C• µi are the 5 intersection points of C with fiber of KS

over each p in S we have 5 points in fiber over S: (p, µi) on C → S

Significance for model building:

The intersection of C and S determines the matter curves


Spectral covers〈φ〉 makes massive components of 248 of E8 other than ad(G) in:

248 → (ad(G), 1) + (1, ad(H)) +∑

i(Ui, Ri)

origin: interaction [〈φ〉, δφ]2 ↔ mass terms of Ui given by weights λi(Ri)

Example : 248 7→ (24, 1) + (1,24) + [(10,5) + (5,10) + h.c.]

• Generically on S: only (24, 1) massless, all others massive

• mass for (10,5) ↔λi weights for fundamental 5 of H

• Relation to roots:

λ1 = α4, λ2 = α3 + α4, λ3 = α2 + α3 + α4,

λ4 = α1 + α2 + α3 + α4, λ5 = α−θ + α1 + α2 + α3 + α4

• if λi = 0 for some i = 1, . . . 5: extra P1 shrinks

→ extra massless 10i from (10,5) ⊂ E8

• location of full 10 matter curve on S:∏

i λi = 0


Spectral coversRelation to Tate picture

y2 = x3 + b0w5 + b2xw3 + b3yw2 + b4x

2w + b5xy :

SO(10) enhancement along b5 = 0 ⇔ Identify b5 ≃∏i λi

Tate algorithm correctly reproduced if one identifies

bn ≃ b0en(λi) → µi ≡ λi,

In particular: b1 =∑

i λi ≡ 0 for SU(n)

Non-trivial check: 5 curve ↔ weights λi + λj for 10 of SU(5)⊥

Full 5 curve P5:∏

i<j(λi + λj) = 0

together with b1 = 0 ⇒ b23b4 − b2b3b5 + b0b

25 = 0 X

fundamental weights λi = eigenvalues µi of Higgs field

Note: For gauge groups other than SU(n) Casimirs en replaced by

corresponding group theoretic invariants


Spectral covers: FluxActual power of spectral cover: description of gauge flux G

2 types of flux:

• F2 on S with values in G = SU(5) ↔ G4 ≃ F2 ∧ ωGi ,

F2 ∈ H(1,1)(S, Q) ⇒ breaks gauge group on S

• F2 on extra branes constituting I1 locus

leaves G on S unaffected and allows for chiral matter

Locally, the I1 locus is described by spectral cover C(5)

⇒ flux on I1 ↔ line bundle N on C(5): c1(N ) ∈ H(1,1)(C(5), Z)

(1,1)-form on C(5) ↔ G4 = F2 ∧ ωHi F2 ∈ H(1,1)(S, Z)

• Within X, fibration structure of C(5) → S with fiber f = µi implies:

H2(C(5), Z) = H2(S, H0(f))

• on corresponding ALE fibration: H0(f) → 〈ωHi 〉


Spectral covers: FluxWhy is I1 flux described as G4 = F2 ∧ ωH

i with F2 ∈ H(1,1)(S, Z)?

• Spectral cover is local ”first order”description of geometry of S and

the neighbouring I1 locus

• 2 ways to break original E8 symmetry:

by gauge flux and by scalar deformations

• E8 symmetry ↔ all branes along same locus S

spectral cover C(5) = product of tilting a suitable number of copies of

branes on S

locally this is the I1 locus, i.e. the remaining branes in the system

• flux on I1 at same order of deformation as gauge flux in representation

of H, but with 2 legs on S

2 legs along I1 and in H would be ”second”order

at least for all purposes of chiral matter on curves entirely S correct

description of I1-flux


Spectral coversExample G = SU(5):

for generic spectral cover H = SU(5)

more generally:

H can contain abelian factors

• C(5) splits into 2 or more connected components

• In Tate model: I1 splits - at least in neighbourhood of S!

Example:

H = S[U(4) × U(1)] ↔ split spectral cover C5 = C4 ∪ C1

Will become relevant for model building:

1) More flexibility for flux solutions

2) resulting U(1) may forbid unwanted couplings


Global aspectsHow exact is the spectral cover construction (SCC)?

• A priori designed to capture only local neighbourhood of S within B

• based on a local P1 fibration over S: expect correct description of

matter curves and Yukawas on S

To some extent, SCC captures also global information

• For models with heterotic dual, SCC allows us to compute χ24 exactly

Method: Comparison of D3-tad with M5-tad in heterotic M-theory

• This formula turns out correct for global Tate models also in cases

without heterotic dual!

Blumenhagen,Jurke,Grimm,T.W.; Grimm,Krause,T.W. ’09

However: SCC is blind to physics away from non-abelian brane locus!

In particular: Insensitive to global existence of massive or massless U(1)

gauge groups


Appendix: χ from SCCIngredients of heterotic E8 × E8 on elliptic Z:

• VEV to internal curvature Fij in subgroup H1 × H2 ⊂ E8 × E8

↔ holomorphic vector bundle V1 × V2 with structure group H1 × H2

gauge group: E8 × E8 → G1 × G2, Gi = E8/Hi

3-form H3 = dB2 + ωCS(F ) − ωCS(F ): dH3 = α′

4 (trF 2 − trR2)

• M5 branes along 2-cycle γ with dual 4-form Γ

Tadpole condition:∑

i Ni[Γi] = c2(Z) + ch2(V1) + ch2(V2)

Special case: M5 wraps T 2 fiber of Z → Γ supported on B2:

→ NM5 =

∫

B2

c2(Z) + ch2(V1) + ch2(V2)

• c2(Z) = 12σc1(B2) + 11c21(B2) + c2(B2)

• ch2(Vi) depends on concrete bundle


Appendix: χ from SCCOn elliptic CY Z a class of such bundles can be described via SCC

For structure group Hi = SU(Ni):

• [Ci] = niσ + π∗(ηi), line bundle N•

∫

B2c2(V1) =

∫

B2η1σ − 1

24χSU(n) − 12

∫

B2πn∗(γ

2)

χSU(n) =∫

Sc21(S)(n3 − n) + 3n η

(η − nc1(S)

)↔ info of [Ci]

∫

B2πn∗(γ

2) ↔ information of line bundle c1(N )

• ⇒ η(1) = 6c1(B2) − t η(2) = 6c1(B2) + t from σ part in N5

Special case: E(2)8 broken completely via H2 = E8

χ(2)E8

= 120∫

S

(3η2

(2) − 27ηc1(S) + 62c21(S)

) ∫

B2πn∗(γ

2) = 0

N5 =

∫

B2

(11c2

1(B2) + c2(B2))

+1

24

(

χSU(1)(n) + χE

(2)8

)

+1

2

∫

B2

πn∗(γ2)


Appendix: χ from SCCDual F-theory model:

• gauge group G1 along S t = c1(NS/B)

• M5-brane along T 2 −→ D3 brane at point on B

• N5 = N3 = χ(Y )24 − 1

2

∫

YG ∧ G

χ(Y )24 =

∫

S

(11c2

1(B2) + c2(B2))

+ 124

(

χSU(1)(n) + χE

(2)8

)

Important observation:

for H1 = E8 no non-ab. gauge group along S

→ Y is smooth and χ(Y )∗ = 360∫

Bc31(B) + 12

∫

Bc1(B) c2(B)

Eliminate∫

S

(11c2

1(B2) + c2(B2))

and find:

χ(Y) = χ∗(Y) + χSU(1)(n) − χE

(1)8

This formula holds true even in absence of heterotic dual!


Overview

Part III:

F-theory GUT phenomenology

• SU(5) GUTS: basics

• GUT symmetry breaking

• Proton decay

• U(1) restricted Tate model

• Summary


F-theory GUTs

Aim: construction of 4D models with realis-

tic particle phenomenology

Guideline: unification of gauge couplings at

MX ≃ 1016GeV

Strategy: Localisation of gauge sector to ex-

plain hierarchy MGUT = 10−3MPl.pic: Kazakov, 0012288

• S10D = M8∗

∫

R1,3×B3

√−gR → M2Pl. = M8

∗ Vol(B3) M∗ = ℓ−1s

• SY M = M4∗

∫

R1,3×SF 2 → α−1

GUT = M4∗ Vol(S)

• GUT breaking (see later) → M4GUT = Vol(S)−1

3 different scales Vol(S), Vol(B3), M∗ to achieve both

• MPl. = 103MGUT = 1019GeV

• α−1GUT = 24


F-theory GUTsAchieving the hierarchy:

• In principle enough to ensure mild tuning ℓs︸︷︷︸

0.2x

< RS︸︷︷︸

2.2x

< RB3︸︷︷︸

5.6x

x = 10−16GeV−1

• Eventually this has to happen dynamically!

• Stronger requirement: decoupling MPl. → ∞ while MGUT finite

While physically not absolutely necessary, appears as a sensible

organising principle of GUT model building BHV & DW ’08

• Mathematical theorem:

decoupling forces S to be Fano∫

ΓK−1

S > 0 ∀ 2-cycles Γ

• 2 kompl.-dim. Fanos: P1 × P1, P2, del Pezzo dPr, r = 1, . . . , 8

→ severely constrains possible GUT surfaces

• for del Pezzo H(0,2)(S) = 0 = H(0,1)(S)

⇒ no deformation modes or continuous Wilson lines

Welcome feature: avoiding adjoint exotics would require stabilisationTor Vergata Roma, November 2010 – p.61

F-theory GUTsdel Pezzo convenient choice, but - unfortunately - there is no

general prediction that S must be of that type

Still: Assume now that S is del Pezzo


F-theory GUTs• Simplest realisation: SU(5) GUTs BHV & DW ’08

• SO(10) possible, but exotics arise in process of GUT breaking

Aim: SU(5) GUT group along divisor S ⊂ B3 given by locus w = 0

→ y2 = x3 + f(ui, w)xz4 + g(ui, w)z6 with f(ui, w), g(ui, w) s.t.

• ∆ = w5P (ui, w),

• P (ui, w) 6= w p(ui, w)

=⇒ SU(5)︸︷︷︸

S:w=0

× U(1)︸︷︷︸

D:P (ui,w)=0

a) matter:

enhancement of singularity type by rank one on intersection locus S ∩ D

• SU(5) × U(1) → SU(6)

35 → 24 + 1 + 5 + 5 =⇒ 5m = (dcR, L) or 5H + 5H

• SU(5) × U(1) → SO(10)

45 → 24 + 1 + 10 + 10 =⇒ 10 = (QL, ucR, ec

R)

N cR: any SU(5) singlet with suitable couplings


F-theory GUTsb) Yukawa couplings

further enhancements at mutual intersections of curves

• 105m 5H for masses of down quarks

• 10105H for masses of up quarks

• 5H 5m 1Nc

Rfor Dirac neutrino masses

Group theory:

• 〈1055〉 ⊂ 〈(66)3〉 of SO(12) =⇒ enhancement SU(5) → SO(12)

possible also perturbativey in IIB theory

• 〈10105〉 ⊂ 〈(78)3〉 of E6 =⇒ requires exceptional enhancements

not possible perturbatively in IIB theory

(however: D-brane instantons can do it

genuine F-theory effect

• 〈551〉: SU(7) point sufficient


SU(5) GUT breaking3 ways to break SU(5) → SU(3) × SU(2) × U(1)Y

• via GUT Higgs in adjoint 24 of SU(5) ↔ brane moduli H(0,2)(S)

problem: explicit GUT breaking potential from string theory ↔stabilisation of brane moduli?

Note: Fluxes do stabilise moduli, but hard to arrange suitable potential

• Wilson lines ↔ brane moduli H(0,1)(S)

again challenge of stabilising open moduli:

now fluxes not even avaliable

viable special case: discrete Wilson lines from discrete π1(S)

• by line bundle LY ↔ hypercharge U(1)Y

works for divisors with H(0,1)(S) = 0 = H(0,2)(S)

↔ open moduli problem avoided → Xdel Pezzo S

We take U(1)Y - approach:

first applied in F-theory context by [Beasley,Heckman,Vafa ’08]

same as in heterotic: [Blumenhagen,Honecker,TW ’06]


SU(5) GUT breaking (II)Idea: VEV to FY on S with TY = diag(−2,−2,−2, 3, 3) ⊂ SU(5)

SU(5) −→ SU(3) × SU(2) × U(1)Y

24 → (8,1)0Y+ (1,3)0Y

+ (1,1)0Y+ (3,2)5Y

+ (3,2)−5Y

5 → (3,1)2Y+ (1,2)−3Y

10 → (3,2)1Y+ (3,1)−4Y

+ (1,1)6Y,

5H → (3,1)−2Y+ (1,2)3Y

, 5H → (3,1)2Y+ (1,2)−3Y

3 challenges:

1) U(1)Y must remain massless X

2) no exotic states (3,2)5Y+ (3,2)−5Y

from 24 X

3) preservation of unification ???


SU(5) GUT breaking (III)Massless U(1)Y

Generically an abelian gauge factor acquires mass via Stuckelberg

mechanism:

C(2) ∧ F -type coupling in 4DU(1)c

Analyse in IIB 7-brane language:

SCS =∫

IR1,3×S

(ι∗C8 + trF ∧ ι∗C6 + tr(F 2) ∧ ι∗C4 + . . .

)

Expand: F = TY

(

F 4DY + c1(LY )

)

+ (other SU(5) generators)

2 potential sources for mixing term:

• ι∗C6 =∑

a C(a)2 ∧ ι∗αa, αa ∈ H4(Y )

• ι∗C4 =∑

i C(i)2 ∧ ι∗ωi, ωi ∈ H2(Y )

since trTY = 0: only term involving tr(F 2) relevant

⇒ F 4DY ∧ C

(i)2 trT 2

Y

∫

S(c1(LY ) ∧ ι∗ωi)

⇒ U(1)Y massless iff c1(LY ) orthogonal to ι∗H2(B, Z)


SU(5) GUT breaking (IV)Important concept: gauge flux in relative cohomology of S ⊂ B3

gauge flux on divisor S ⇔ H2(S)

2 types of 2-forms on divisor S: Lerche, Mayr, Warner ’01/02;

pullbacks from B vs. 2-forms not inherited from B3 Jockers,Louis’05

The latter 2-forms are dual to 2-cycles in H2(S) which are a boundary of a

3-chain in B

If c1(LY ) only through such ”relative”2-cycles on S, then∫

S(LY ∧ ι∗ωi) = 0

Constraint: hypercharge flux must lie in relative cohomology of S in B3

Non-trivial constraint on compactification geometry:

rigid GUT divisor must allow for such gauge flux which is not pull-back

from ambient space

global feature - depends on compactification details, not on local data!


MSSM matterNecessary condition: Recall: l2 = 1, Ei · Ej = −δij

if S is del Pezzo, then c1(S) = 3l −∑i Ei inherited from B3 →c1(LY ) · c1(S) = 0 ↔ c1(S) in Weyl lattice 〈l − E1 − E2 − E3, Ei − Ej〉Absence of exotics

24 → (8,1)0Y+ (1,3)0Y

+ (1,1)0Y+ (3,2)5Y

+ (3,2)−5Y

• localised on entire divisor S

• counted by combinations of Hi(S,L5Y ) = 0

for c1(LY ) in Weyl lattice of dPr, this is not possible:

Hi(Ei − Ei) = 0, but not for 5th power!

But can redefine embedding of LY such that Hi(S,L±1Y ) appears

In SCC we also have the SU(5)⊥ ⊂ E8 bundle V on S

Define S[U(5) × U(1)Y ] bundle V ⊕ LY : c1(V ) + c1(LY ) = 0

Cartan generators of the structure group of V ⊕ LY :

(1, 1, 1, 1, 1︸︷︷︸

V

, −5︸︷︷︸

LY

) ⊂ SU(6) ⊂ E8


MSSM matter

248 7→

(24;1,1)0 + (1;1,1)0 + (1;8,1)0 + (1;1,3)0

(5;3,2)1 + (1;3,2)5 + c.c.

(10;3,1)2 + (5;3,1)−4 + c.c.

(10;1,2)−3 + (5;1,1)6 + c.c.

SU(3) × SU(2) × U(1)Y bundle Standard Model particles

(3,2)1 V qL L-handed quark

(3,2)5 L−1Y − (exotic matter)

(3,1)2∧2

V dL = dcR L-handed down antiquark

(3,1)−4 V ⊗ LY uL = ucR L-handed up antiquark

(1,2)−3

∧2V ⊗ LY lL L-handed lepton

(1,1)6 V ⊗ L−1Y eL = ec

R L-handed antielectronTor Vergata Roma, November 2010 – p.70

MSSM matter - Higgsappearance of full GUT matter multiplets:

c1(LY ) · P10 = 0 = c1(LY ) · P5m

Higgs sector:

5H → (3,1)−2Y+ (1,2)3Y

, 5H → (3,1)2Y+ (1,2)−3Y

controlled mechanism for stringy doublet-triplet splitting via U(1)Y

[Beasley,Heckman,Vafa ’08]

PH splits into Pu ∪ Pd such that 5H along Pu, 5H along Pd

net U(1)Y flux through Pu and Pd such that:

H∗(Pu,∧V ⊗ LY ) = (1, 0), H∗(Pd,∧V ⊗ LY ) = (0, 1)

↔ one Hu and Hd

H∗(Pu,∧V ) = (0, 0), H∗(Pd,∧V ) = (0, 0)

↔ massless triplets projected out

topological solution to doublet triplet splitting


Proton DecayConventional SU(5) GUTs suffer from too large proton decay

Dimension 4:

• R-parity violating terms ucR dc

R dcR , L L ec

R, Q L dcR ↔ 105m 5m

• must be absent due to experimental bound

• Necessary condition: Pm, PH on different curves

otherwise: 105m 5H implies 105m 5m

• Note: This is not sufficient - see later

Dimension 6:

• From exchange of heavy gauge bosons in (3,2) + (3,2)

• leads to decay of type p → π0 + e+

• decay rate is well within experimental bounds → no problem X


Proton decayDimension 5:

• effective terms of type c2

Meff(QQQL + uc

RucRdc

RecR)

• lead to decay of type p → K+ν

• strongly constrained by proton lifetime

• Generation via triplet exchange: 5H = (Tu, Hu),5H = (Td, Hd):

QQTu + QLTd + MKKTuTd → 1MKK

Q Q Q L

• realised if triplets acquire mass by pairing up with each other

↔ Hu, Hd on same curve PH

• Remedy: Missing partner mechanism:

QQTu + QLTd + MKKTuTd + MKK TuTd

=⇒ no integration to QQQL possible

• Necessary: need Hu and Hd on two separate curves

• Note: In this case there is also no O(1) µ term µHuHd X


SU(5) GUTs

pic from: Beasely,Heckman,Vafa, 0806.0102


U(1) selection rules• Generic Tate model: single 5-matter curve

=⇒ 105m 5H ↔ 105m 5m: dimension 4 proton decay

• Remedy: split P5 → Pm + PH + effective U(1) selection rule

[Beasley,Heckman,Vafa II ’08] [Hayashi,Kawano,Tatar,Watari 0901.4941]

Group theory: E8 → G × H, G = SU(5)GUT

• H = SU(5)⊥ → S[U(4) × U(1)X ]

• U(1)X charges: 101 (5m)−3 (5H)−2 + (5H)2 1−5

• If U(1)X unbroken in fully fledged model, then 105m 5m forbidden X

• Necessary condition for U(1)X : split P5 → Pm + PH

achieved by split spectral cover → realises S[U(4) × U(1)X ] ”locally”

[Marsano,Saulina,Schafer-Nameki 0906.4672][Tatar,Tsuchiya,Watari 0905.2289]


Global caveatsApplied to construction of 3-generation models:

[Marsano,Saulina,Schafer-Nameki 0906.4672][Blumenhagen,Grimm,Jurke,TW 0908.1784]

[Marsano,Saulina,Schafer-Nameki 0912.0272] [Grimm,Krause,TW 0912.3524]

[Chen,Knapp,Kreuzer,Mayrhofer 1005.5735]

Caveat:

• U(1)X might be higgsed by GUT singlets Φ

[Tatar,Tsuchiya,Watari 0905.2289],[Grimm, TW 1006.0226]

• happens away from GUT brane ⇐⇒ beyond regime of spectral cover

• This is a global question of direct physical relevance:

If U(1)X higgsed, effective proton decay operators generated

W ⊃ 1

M105m 5m Φ =⇒ 〈Φ〉

M105m 5m

Independent analysis via monodromies: [Hayashi,Kawano,Tsuchiya,Watari

1004.3870]


U(1) restricted Tate modelConstructive method to ensure presence of abelian gauge symmetries:

[Grimm, TW 1006.0226]

• In generic models U(1) absent due to maximal Higgsing compatible

with non-abelian gauge symmetries ↔ VEV for gauge singlets 〈1−5〉• U(1) symmetries recovered by unhiggsing

→ massless gauge singlets away from GUT brane

• requires singular matter curves away

from GUT brane

• self-intersection of I1 locus

enhancement I1 → A1 ≃ SU(2)

• further specification of complex

structure:

U(1) restricted Tate model


U(1) restricted Tate modelSafe way to check for U(1)X : [Grimm,TW 1006.0226]

Resolve space YG → Y G and count h1,1(Y G)

Result: nU(1) = h1,1(Y G) − h1,1(B) − 1 − rk(G) = 1

Reason: Resolution divisor DC for singular curve C ↔ dual 2-form ωC

C3 = AX ∧ ωX +∑

i

Ai ∧ ωGi + . . . ωX ↔ ωC

• presence of U(1) does not hinge on any factorisation of the

discriminant

• Compatible with appearance of U(1) symmetries in IIB limit

F-theory lift does not destroy U(1) - only affects split

O7-plane/D7-brane intersection

• U(1) are non-generic both in IIB and in F-theory


U(1) restricted Tate modelPractical consequence of U(1) restriction: χ(Y ) decreases drastically

GUT example of [Grimm,Krause,TW 0912.3524]

Generic SU(5) Tate model: χ = 5718 ↔ U(1)-restricted model: χ = 2556

What about U(1)X flux?

• bundle is now of type S[U(4) × U(1)X ]

• global description in terms of G4 complicated

tempting: G = FX ∧ ωX + . . . ? details still under investigation

• U(1)X flux =⇒ Fayet-Iliopoulos D-term

↔ U(1)X acquires Stuckelberg mass in presence of suitable gauge flux

=⇒ global selection rule ↔ instantons → work in progress


Summary of approachCharacteristic features of F-theory model building:

• localisation of gauge d.o.f.

• emergence of exceptional gauge groups

=⇒ good for GUTs

We have seen interesting mechanisms to accomodate:

• all required Yukawas - in principle X

• GUT breaking without GUT Higgs X

• suppress proton decay X

• achieve doublet-triplet splitting X

Many further phenomenological directions have been explored, inlcuding:

• hierarchy in flavour sector/structure of Yukawa couplings

• neutrino sectorsTor Vergata Roma, November 2010 – p.80

Summary of approachWhat about explicit constructions?

Many of the above realised in explicit compact models - thanks to

• spectral cover construction

• detailed understanding of Calabi-Yau fourfolds

Urgent directions:

• no stabilisation of moduli achieved

• no dynamical argument for specific choice of brane moduli

• coupling to, say, inflationary cosmology...


F-theory compactiﬁcations and model buildingstatistics.roma2.infn.it/~stringhe/doc/weigand.pdf ·...

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