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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)
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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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Grand Unification from F-theory
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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Grand Unification from F-theory
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Grand Unification from F-theory
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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Gauge coupling unification
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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Gauge coupling unification
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Gauge coupling unification
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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Gauge coupling unification
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Gauge coupling unification
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!