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F -theory GUTs
J. Marsano
Enrico Fermi Institute
University of Chicago
In collaboration with: N. Saulina, S. Schäfer-Nameki
0808.1286
,0808.2450
,0904.3932
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Particle Physics and String Theory
• How can we hope to say anything about particle physics giventhe complexity of string vacua?
• A hint from nature that we are in a good situation. . .
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Particle Physics and String Theory
• How can we hope to say anything about particle physics giventhe complexity of string vacua?
• A hint from nature that we are in a good situation. . .
• Apparent unification ofcouplings in MSSM atM GUT ∼ 1016 GeV
→ M SUSY ≪ M GUT ≪ M Planck
14 15 16 17LogM 1 GeV
24
26
28
30
32
1Αr
t
• Suggests that particle physics captured by SUSY gauge theorythat isn’t too sensitive to details of quantum gravity
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Bottom-up Approach
[Aldazabal, Ibanez, Quevedo, Uranga], [Gray, He, Jejjala, Nelson][Verlinde, Wijnholt]
• Look for framework with natural separation between gauge andgravity degrees of freedom
→ Type II models with branes
1. Smooth compactification with intersecting branes (this talk)
[Cvetic, Shiu, Uranga], + many others
2. Branes probing singular compactifications
[Aldazabal, Ibanez, Queveco, Uranga], [Verlinde, Wijnholt], + many others
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Charged Matter
SU(5)
U(1)
SU(6) Bifundamental • Charged matter from openstrings with one end on thestack
• Other end on some other
D-brane, orientifold plane,→ "Matter branes"
• Can be described by larger rank gauge theory with nontrivialadjoint vev
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Worldvolume Gauge Theory
• Study worldvolume gauge theory with gauge group G ⊃ SU (5)
• Nontrivial vev for adjoint scalar field
φAdj = 0 → G → SU (5) {×U (1)m }
• Large brane rotations → string scale vevs
• Nevertheless can compute holomorphic data
• Spectrum• Superpotential
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MSSM from SU (5) GUT
• Supersymmetric SU (5)GUT GUT
3×10M ∼
Q ∼ (3, 2)+1/6
U c ∼ (3, 1)−2/3
E c ∼ (1, 1)+1
3×5M ∼
D c ∼ (3, 1)+1/3
L ∼ (1, 2)−1/2
5H ∼
H u ∼ (1, 2)+1/2
H (3)u ∼ (3, 1)−1/3
5H ∼
H d ∼ (1, 2)−1/2
H (3)d ∼ (3, 1)+1/3
W ⊃ λUP10M × 10M × 5H , λDOWN10M × 5M × 5H
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Which G ’s are Possible?
G → SU (5)×U (1)
k Adj(G ) → Adj(SU (5))⊕
Adj(U (1)k
⊕ [Bifundamentals]
• MSSM matter
10 and 5 ∈ Bifundamentals
• MSSM Superpotential
10× 10× 5, 10× 5× 5 ⊂ Adj(G )3
=⇒ G ⊃ E 7
Forced to nonperturbative realm of type II theories
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F-theory and M-theory
M -Theory Models
• Intersecting "6-brane" models
• 7-dimensional gauge theory
on a R 3-manifold X 3[Pantev, Wijnholt]
• Charged matter localized atpoints in X 3
• Yukawas exponentiallysuppressed but no candidatemodel of flavor
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F-theory and M-theory
M -Theory Models
• Intersecting "6-brane" models
• 7-dimensional gauge theory
on a R 3-manifold X 3[Pantev, Wijnholt]
• Charged matter localized atpoints in X 3
• Yukawas exponentiallysuppressed but no candidatemodel of flavor
F -Theory Models
• Intersecting "7-brane" models
• GUT → 8-dimensional gauge
theory on a C surface S [Donagi, Wijnholt]
[Beasley, Heckman, Vafa]
• Charged matter localized oncurves in S
• Some proposed mechanismsfor flavor structure
[Heckman, Vafa]
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F-theory and M-theory
M -Theory Models
• Intersecting "6-brane" models
• 7-dimensional gauge theory
on a R 3-manifold X 3[Pantev, Wijnholt]
• Charged matter localized atpoints in X 3
• Yukawas exponentiallysuppressed but no candidatemodel of flavor
• Difficult to study
F -Theory Models
• Intersecting "7-brane" models
• GUT → 8-dimensional gaugetheory on a C surface S
[Donagi, Wijnholt][Beasley, Heckman, Vafa]
• Charged matter localized oncurves in S
• Some proposed mechanismsfor flavor structure
[Heckman, Vafa]
• Simpler to study
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Focus
• We will study GUTs in F -theory, namely
8-dimensional E 8 gauge theory on a C surface S
• Will now see how much progress we can make towards
• Breaking GUT Group
• Suppressing Proton Decay
• Addressing Unification Issues
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Aside: Connection to Heterotic
[Vafa], [Morrison, Vafa], [Friedman, Morgan, Witten][Curio, Donagi], [Donagi, Wijnholt], [Hayashi, Tatar, Toda, Watari, Yamazaki]
• 8d E 8 gauge theory → SU (5) with φadj = 0
• Field configuration for φAdj must satisfy certain BPS equations
• Solutions can be constructed using spectral cover
• Near GUT 7-branes, geometry a local K3-fibration over S
• Can embed into global K3 fibration with a Heterotic dual
• Geometric deformation↔ E 8 spectral bundle
• Constructed from spectral cover
Gauge Theory Description ↔ "Heterotic Description"
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Charged Matter
SU(5)
U(1)
SU(6) Bifundamental
SU(5)
S
5
"Matter Curve"
GUT
Charged matter is effectively 6-dimensional
• Spectrum of 4d multiplets requires further dimensional reduction• # of 4d multiplets can be adjusted with worldvolume fluxes
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GUT-Breaking
SU (5) → SU (3) × SU (2)×U (1)Y
24 → (8, 1)0 ⊕ (1, 3)0 ⊕ (1, 1)0 ⊕
(3, 2)−5/6 ⊕ cc
• Must break SU (5) and project out the lepto-quarks• Promising idea: Internal U (1)Y flux on S
[Beasley, Heckman, Vafa] [Donagi, Wijnholt]
• Can lift lepto-quarks as well as Higgs triplets
H ∼
H u ∼ (1,2)+1/2
H (3)u ∼ (3, 1)
−1/3
ffH ∼
H d ∼ (1,2)
−1/2
H (3)d ∼ (3,1)+1/3
ff
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Dimension 4 Proton Decay
• Must forbid the dangerous operator
W Proton Decay ∼ 10M × 5M × 5M
• How to do it?
• Discrete symmetries – Difficult in practice[Tatar, Tsuchiya, Watari]
• Continuous symmetries – plentiful[JM, Saulina, Schäfer-Nameki]
E 8 → SU (5)× U (1)4
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Which Symmetries?
W MSSM ∼ 10M × 10M × 5H + 10M × 5M × 5H
+10M × 5M × 5M + 10M × 5H × 5H
• MSSM Superpotential Preserves a 1-parameter family of U (1)’s
10M 5M 5H 5H
U (1) 1 q − 3 −2 2− q
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Which Symmetries?
W MSSM ∼ 10M × 10M × 5H + 10M × 5M × 5H
+10M × 5M × 5M + 10M × 5H × 5H
• MSSM Superpotential Preserves a 1-parameter family of U (1)’s
10M 5M 5H 5H
U (1) 1 q − 3 −2 2− q
• When q = 0, Q ∼ Q Y + Q B −L
• Gauged B − L models
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Which Symmetries?
W MSSM ∼ 10M × 10M × 5H + 10M × 5M × 5H
+10M × 5M × 5M + 10M × 5H × 5H
• MSSM Superpotential Preserves a 1-parameter family of U (1)’s
10M 5M 5H 5H
U (1) 1 q − 3 −2 2− q
• When q = 0, Q ∼ Q Y + Q B −L
• Gauged B − L models• q = 0 → PQ symmetry
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Dimension 5 Proton Decay
• There are tight bounds on the operator responsible for dimension
5 proton decay1
M
d 2θQ 3L ⊂
1
M
d 2θ
10M × 10M × 10M × 5M
• Mechanisms like "missing partner" have been proposed in
F -theory models before Beasley, Heckman, Vafa
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Dimension 5 Proton Decay
• There are tight bounds on the operator responsible for dimension5 proton decay
1
M
d 2θQ 3L ⊂
1
M
d 2θ
10M × 10M × 10M × 5M
• Mechanisms like "missing partner" have been proposed in
F -theory models before Beasley, Heckman, Vafa
• Unfortunately, KK modes will generate any allowed operatorsJM, Saulina, Schäfer-Nameki
10M 5M 5H 5H
U (1) 1 q − 3 −2 2− q
• The above operator carries U (1) charge q
→ Allowed if q = 0
=⇒ We must realize U (1) with q = 0 – need U (1)PQ symmetry
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Comments on U (1)PQ Symmetries
• Prevent a bare µ term
W µ ∼ µ 5H × 5H
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Comments on U (1)PQ Symmetries
• Prevent a bare µ term
W µ ∼ µ 5H × 5H
• If a PQ -charged field X carries a SUSY-breaking vev, a small µ
term can be generated via Giudice-Masiero type mechanismfrom
d 4θ1
M X † × 5H × 5H
• Possible solution for µ/B µ problem in gauge mediation[Ibe, Kitano], [JM, Saulina, Schäfer-Nameki]
• PQ gauge boson gives additional contributions to SUSY-breakingsoft masses
[Heckman, Vafa]
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Getting the Right Symmetries
• Our U (1)’s come from the parent gauge group,
E 8 ⊃ SU (5)GUT × SU (5)⊥
E 8 → SU (5) × U (1)4 φ ∼
t 1 0 0 0 00 t 2 0 0 00 0 t 3 0 0
0 0 0 t 4 00 0 0 0 t 5
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Getting the Right Symmetries
• Our U (1)’s come from the parent gauge group,
E 8 ⊃ SU (5)GUT × SU (5)⊥
E 8 → SU (5) × U (1)4 φ ∼
t 1 0 0 0 00 t 2 0 0 0
0 0 t 3 0 0
0 0 0 t 4 00 0 0 0 t 5
• φAdj varies over internal space
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Getting the Right Symmetries
• Our U (1)’s come from the parent gauge group,
E 8 ⊃ SU (5)GUT × SU (5)⊥
E 8 → SU (5) × U (1)4 φ ∼
t 1 0 0 0 00 t 2 0 0 00 0 t 3 0 0
0 0 0 t 4 0
0 0 0 0 t 5
• φAdj varies over internal space
• Only gauge invariant quantities, such as trm φAdj, need bewell-defined
→ t i ’s are permuted by a monodromy group G Monod as we move alongthe internal space, S
• G Monod effectively quotients the theory, removing most (or all)
U (1)’s
• G Monod intrinsically connected to "global" structure on S
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Implications of PQ Symmetries
[JM, Saulina, Schäfer-Nameki]
• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much
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Implications of PQ Symmetries
[JM, Saulina, Schäfer-Nameki]
• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much
• Whenever we have a PQ symmetry and
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Implications of PQ Symmetries
[JM, Saulina, Schäfer-Nameki]
• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much
• Whenever we have a PQ symmetry and
1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure
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Implications of PQ Symmetries
[JM, Saulina, Schäfer-Nameki]
• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much
• Whenever we have a PQ symmetry and
1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure
2. GUT-breaking via U (1)Y flux
we also get
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Implications of PQ Symmetries
[JM, Saulina, Schäfer-Nameki]
• Sadly, the rosy scenario we have described doesn’t like PQ symmetries very much
• Whenever we have a PQ symmetry and
1. All 3 generations of MSSM matter localized on a single curve→ for flavor structure
2. GUT-breaking via U (1)Y flux
we also get
3. Extra light non-GUT exotics
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PQ -Symmetry =⇒ Exotic Particles
E ti ?!?!?!?!?
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Exotics?!?!?!?!?
• What to do about exotics?1. Find some other way to get flavor structure
[Ibanez, Font], [Dudas, Palti]
2. Use a different mechanism to break the GUT group
3. Look for exotic structures in string theory beyond E 8
4. Lift the exotics through coupling to a PQ -charged singlet, X
Z d 2θXf Exoticf Exotic
• Reminiscent of gauge mediation
• Interesting phenomenology?
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Exotics are not the only problem with these models
U ifi ti P bl i M d l ith U(1) Fl
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Unification Problem in Models with U (1)Y Flux
• 8-dimensional gauge theory has an important 1-loop divergence
[Donagi, Wijnholt], [Wijnholt]
ln(Λ)
trAdj (F ∧ F ∧ F ∧ F )
• Represents a "local tadpole" that must be canceled globally in a
consistent string compactification[Conlon], [Conlon, Palti]
• Background F Y leads to distortions of 4-d coupling at the KK scale[Blumenhagen], [Wijnholt]
θZ R3,1
F ∧ F → θZ R3,1
F ∧ F + ln„
ΛM KK
«Z R3,1
F ∧ F Z S
F Y ∧ F Y
• Non-universal shifts – splits gauge couplings at the high scale
Comparison with Observation
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Comparison with Observation• Worry about couplings at M Z – this is what we actually measure
α−1i (M Z ) = αGUT −
β (MSSM )i
2πlnM KK
M z
− β (thresh)i
2πln
ΛM KK
• Splittings α−1i (M Z )− α−1
j (M Z ) independent of αGUT butsensititive to threshold contributions• MSSM alone agrees with experiment to within 0.5%
• With correction, we sit on threshold of acceptibility
• Most rosy estimates for Λ can lead to ∼ 0.
5% but most are somewhatlarger
Comparison with Observation
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Comparison with Observation• Worry about couplings at M Z – this is what we actually measure
α−1i (M Z ) = αGUT −
β (MSSM )i
2πln
M KK
M z
−
β (thresh)i
2πln
Λ
M KK
−β (exotic)
2πln
M KK M exotic
• Splittings α−1i (M Z )− α−1
j (M Z ) independent of αGUT butsensititive to threshold contributions
• MSSM alone agrees with experiment to within 0.5%
• With correction, we sit on threshold of acceptibility
• Most rosy estimates for Λ can lead to ∼ 0.
5% but most are somewhatlarger
• Contributions from exotics open up a parameter space of betteragreement but we need
M exotic
> 1013−14 GeV
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Our two problems may be able to "cancel" each other
Moving Beyond the "Charged Sector"
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Moving Beyond the Charged Sector
• Crucial to understand singlet fields that are not localized on theSU (5) branes
1. Right-handed neutrinos are SU (5) singlets
2. Exotic masses come from coupling to SU (5) singlets3. Potential SUSY-breaking spurions?
• Singlets probe geometry away from GUT brane surface, S
• Gauge theory perspective not useful
• Cannot rely so strongly on Heterotic input
Global Fluxes
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Global Fluxes
First problem: Fluxes
• Fluxes that induce chirality constructed essentially throughHeterotic techniques
• Global extension of these fluxes beyond the local neighborhood of
S has always been assumed
• Need intrinsic F -theory understanding of global fluxes in order tostart moving away from the GUT 7-branes
• Some ideas for this. . .
[JM, Saulina, Schäfer-Nameki, in progress]
Compact Models
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Compact Models
• Several compact models do exist (modulo flux assumptions)[JM, Saulina, Schäfer-Nameki] [Blumenhagen, Grimm, Jurke, Krause, Weigand]
• Toy models at best
• Moduli stabilization not addressed
• In known examples, D3-brane tadpole doesn’t leave much room for
extra fluxes for this
N D 3 =χ(Y 4)
24−
1
2
Z Y 4
G ∧G
Z Y 4
G ∧ G ≥ 0
Summary
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Summary
• Unification suggests particle physics embedded in
nonperturbative regime of type II
• "Charged sector" described by 7-d or 8-d gauge theory
• Local geometry has dual Heterotic description that provides
important input
• Structure of 8-d gauge theory (F -theory models) is very rigid
• Serious implications for phenomenology in very large class of
string models
• Moving away from 8-d gauge theory requires new technical
advancements
• A study of 7-d gauge theories relevant for M -theory modelswould be interesting to undertake