Ralph Blumenhagen- The Idea of F-theory GUTs

8/3/2019 Ralph Blumenhagen- The Idea of F-theory GUTs

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The Idea of F-theory GUTs

Ralph Blumenhagen

Max-Planck-Institut fur Physik, Munchen

(Bhg, A. Collinucci, B. Jurke, T. Grimm, T. Weigand)

SUSY2010, Bonn p.1/18


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Grand Unification



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Grand Unification

One-loop running of thethree Standard Modelgauge couplings with

MSSM matter spectrumabove the TeV scale, (El-lis, Kelley, Nanopoulos), (Amaldi,

de Boer, Furstenau), (Langacker,

Luo)

2 4 6 8 10 12 14 16

10

20

30

40

50

60

70

log10()

3

51Y

1

w

1s

Evidence for a supersymmetric Grand Unification at

MX = 2.1 1016

GeV, as for instance: Gauge group: SU(5)

chiral matter in 10 + 5 + 1

Higgs field: 5H + 5H Yukawa couplings: 1010 5H, 1055H, 515H



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Grand Unification from String Theory



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Grand Unification from String Theory

Attempts to realize GUTs from String Theory

Weakly coupled E8 E8 Heterotic String (heteroticorbifolds ) Need large threshold corrections at MX GUT breaking via discrete Wilson lines

F-theory/Type IIB compactifications with (p,q)7-branes: Solves the 10105H Yukawa problem of orientifolds

GUT brane wraps a shrinkable 4-cycle GUT breaking via U(1)Y flux



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F-Theory



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F-Theory

F-theory is a way of book-keeping of the positions of moregeneral (p,q)-7-branes in Type IIB N = 1 compactifications

elliptic fibration : Y B3

Susy Y Calabi-Yau4-fold. Elliptic curve

y2 = x3 +f(u) x+g(u)with complex struc-ture:

= C0 + i e

with j() = 4(24)f3

4f3+27 g2B37-brane

(Vafa,Nucl.Phys.B469:403,1996), (Beasley, Heckman, Vafa,

arXiv:0802.3391+0806.0102), (Donagi, Wijnholt, arXiv:0802.2969)SUSY2010, Bonn p.4/18


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Grand Unification from F-theory



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Working hypothesis: Decoupling of GUT scale from Planckscale localisation of GUT physics on del-Pezzo surfaces(Beasley, Heckman, Vafa, arXiv:0806.0102)

Shortcomings

Missing stringy global consistency conditions: landscapevs. swampland

Physics of abelian gauge symmetries: Green-Schwarzmechanism, Freed-Witten anomalies, (Grimm, Weigand)

Need local mechanism for Susy breaking

gaugemediated susy breaking

closed string Moduli stabilisation, need to explain whysusy breaking is subleading to gauge mediation

series of recent papers: (Beasley, Choi, Donagi, Hayashi, Heckman, Marsano,Saulina, Schafer-Nameki, Vafa, Watari, Wijnholt+ . . . )



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Program:

Embed the local ideas into a global framework: F-theoryon elliptically fibered four-folds with shrinkable 4-cycles

Derivation of the global consistency conditions,

Lift and generalise Type IIB orientifold consistencyconditions to genuine F-theory models

Study of consequences of U(1)Y flux gauge couplingunification

Moduli stabilization via flux and instanton generatedsuperpotentials



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F-Theory



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F-Theory

Gauge symmetry on D: Degeneration of elliptic curve,ADE Kodaira classification

Consistency condition: Degeneration loci can be

described by a compact Calabi-Yau fourfold Y Four-form flux quantisation (chirality)

G4 + 12

c2(Y) Z

D3-tadpole:

ND3 +1

2

Y

G4 G4 =(Y)

24



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Matter fields



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Matter fields

Matter fields aregenerally localised oncurves: C = Da Db

SU(5)

U(1) 5

F-theory: Enhancement of the singularity over theintersection: SU(5) U(1) SU(6)

35 = 240 + 10 + 51 + 51

resp. SU(5)

U(1)

SO(10)45 = 240 + 10 + 102 + 102



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Yukawa couplings



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Yukawa couplings

Yukawa couplings:

SU(5)10

5

5H

The Yukawa couplings which give masses to the MSSM fieldsafter GUT and electroweak symmetry breaking are

10(2,0)

10(2,0)

5H(1,1), 10(2,0) 5

(1,1)5H

(1,1),

1(0,2)

N5

(1,1)5H

(1,1)

Problem: The coupling 10(2,0) 10(2,0) 5H(1,1) is

perturbatively forbidden.Solution: Arises in F-theory from an E6 enhancement of thesingularity.



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GUT symmetry breaking



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GUT symmetry breaking

Symmetry breaking via gauge flux FY: c1(LY) H2(D) has

to be trivial in H2(X), i.e. (LY) = 0.

exotic matter:

24 (8,1)0 + (1,3)0 + (3,2)5 + (3,2)5

i.e. H

(D, L

5

Y) = 0. Solution: One defines fractional line bundles La and LY

via

La = La L

2

5

Y LY = L

1

5

Y



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Compact models



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Compact models

Problem: Realisations of all these local features in genuinecompact F-theory

Study manifolds using methods of toric geometry:

Example: Elliptic fibration over IP3

The fourfold is given by the Weierstrass fibration:

y x z u1 u2 u3 u4 p

q1 3 2 1 0 0 0 0 6q2 0 0 -4 1 1 1 1 0

with Tate constraint

y2 = x3 + xyza1 + x2z2a2 + yz

3a3 + xz4a4 + z

6a6

and an polynomials of degree 4n in u.



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Compact models



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Compact models

roadmap:

Perform transitions of these manifold leading new

del-Pezzo type four-cycles Analyze the new elliptic fibration whether it allows for an

SU(5) GUT with the wanted matter curves and Yukawacouplings tadpole conditions

For chirality turn on extra gauge flux (G4 form flux) andcompute spectra and tadpoles

More details in (Bhg, Grimm, Jurke, Weigand, arXiv:0908.1784), (Grimm, Krause,

Weigand, arXiv:0912.3524), (Marsano, Saulina, Schafer-Nameki, arXiv:0906.4672),

(Chen, Knapp, Kreuzer, Mayrhofer, arXiv:1005.5735)



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Gauge coupling unification



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The gauge couplings at the string/GUT scale are changeddue to the U(1)Y flux (Bhg, arXiv:0812.0248):

fSU(3) = a 12S

Da

c21(La)

fSU(2) = a 1

2

S Da

c21(La) + c21(LY) + 2c1(LY) c1(La) 3

5fU(1)Y = a

1

2S

Da

c21(La) +3

5

c21(LY) + 2c1(LY) c1(La)

,

with a = e12

Da

J J. The MSSM gauge couplings

satisfy the relation

1Y(Ms)

= 1w(Ms)

+ 23 s(Ms)

.



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Include the Higgs triplet above a scale 1TeV < M33 < MX inthe running:

(b3, b2, b1) = (3,1,11) (b3, b2, b1) = (2,1,35

3) .

Choosing for instance M33 = 1015

GeV, the running aroundthe GUT scale changes as

14.0 14.5 15.0 15.5 16.0 16.5 17.0

22

24

26

28

30

log10()

3

51Y

1w

1s

M33



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zooming further in

16.1 16.2 16.3 16.4 16.5

24.0

24.5

25.0

log10()

3

51Y

1

w

1s

MX

The three MSSM gauge couplings satisfy the F-theory GUTrelation at

MX = 2.1 1016 GeV

independent of the triplet mass scale M33.



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Moduli stabilization



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Moduli stabilization

Uplift of KKLT and LARGE volume scenario from Type IIB toF-theory

G3 flux becomes G4 form flux with one leg on T2

D3-brane instantons M5-brane instantons, (Bhg,Collinucci, Jurke, arXiv:1002.1894)

technically challenging: dilaton varies over background

New scenario: gravity mediated susy breaking on shrinkableMSSM 4-cycle

Soft masses are suppressed relative to the gravitino

mass: Msoft =M3/2Vp

, p = 1/2, 1 (Bhg, Conlon, Krippendorf, Moster,Quevedo, arXiv:0906.3297)



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Conclusions



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Conclusions

F-theory models can provide string theory realisations ofSU(5) GUTs on localized branes.

Gauge fields arise from degenerations of elliptic fiberover surfaces

Matter fields and Yukawa couplings arise from singularityenhancements over curves and points

Global compact models showing many of the desiredfeatures can be constructed

GUT breaking works via U(1)Y

flux problemswithgauge coupling unification

Determination of flavour structure and Yukawahierarchies by subleading corrections devil in thedetails



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Further progress both on the technical andthe phenomenological level is expected

Thank you!

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Ralph Blumenhagen- The Idea of F-theory GUTs

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