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Moduli fixing, flux vacua and landscape analysis
Lecture 1: Overview of challenges and approaches
Frederik Denef
KU Leuven
Tehran, April 13, 2007
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Outline
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The Landscape Problem
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Challenges for string phenomenology
L
LHC
S
?
1. Fundamental mass scale Ms � Mew
2. Controlled vacua ↔ realistic vacua
3. Multitude of vacua
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Controlled vacua ↔ realistic vacua: susy moduli
R1,3
X6
I String vacua ≡ solutions ∼ R1,3 × X6 to (corrected) 10dsupergravity.
I Susy most efficient to control corrections.
I But: susy solutions (e.g. type II with X6 CY) come with
geometric moduli massless scalars. ×I Quantum corrections can lift those, but...
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Controlled vacua ↔ realistic vacua: weak coupling runaways
I Modulus ρ = Vol(X6) or 1/g2s or eVol(C)/gs or ...
I Example potential:
V (ρ) =1
ρ2− 1
ρ3+
1
4ρ4+O(
1
ρ5).
1 2 3 4 5
0.1
0.2
0.3
0.4
V
ρweakstrong
?
I Weak coupling ρ→∞ = flat 10d space ⇒ V (ρ) → 0.×I Minima “naturally” at strong coupling, beyond control...
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Fluxes
= p-form field strengths F = Fµ1···µpdxµ1 ∧ · · · ∧ dxµp .
Nontrivial p-cycles Ci in X6 can carry quantized magnetic F -flux:∫Ci
F = Ni ∈ Z
Energy density in effective 4d theory:
V (φ,N) = V0(φ) +
∫X6(φ)
|F |2 = V0(φ) + gij(φ)N iN j
For suitable N, stabilized, reasonably controlled vacua! [KKLT]
But...
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Multitude of vacua
Large degeneracy from fluxes
Large degeneracyfrom choice of topology
Large degeneracyfrom moduli potential
Low energy effective field theory parameters
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How big is the set of string vacua?
I infinite: [AdS5 × S5]N ' N = 4 SU(N) SYM, N ∈ Z.I but, surprisingly, if rough observational constraints + control
are imposed, all evidence points (nontrivially) to finitenumber. E.g.:
I A priori infinitely many compactification topologies, butCheeger’s finiteness theorem “⇒” only finite set hasKaluza-Klein radii bounded above while keeping corrections togeometry under control. [Acharya-Douglas]
I # susy flux vacua for given X6 easily ∼ 10500, butI None of these 10100n-type models fully established as
metastable susy-breaking vacua (in KKLT scenario, largenumber of cycles typically leads to sub-string-size cycles henceloss of control).
I Not even roughly known how many vacua there are compatiblewith present experimental data (might even be zero!)
I So, perhaps we can just figure out our vacuum from exp data?
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Low energy predictions from string theory in four easy steps
1. Construct/enumerate all vacua meeting rough observationalconstraints (4 huge dim, no massless scalars, tdecay > 10 Gyr, ...).(labeled by discrete microscopic data ~m: topology, flux, criticalpoints, ...)
Table 6 – continued from previous page
nr Total occ. MIPFs Chan-Paton Group Spectrum x Solved
411 31000 17 U(3)× U(2)× U(1)× U(1) AAVA 0 Y
417 30396 26 U(3)× U(2)× U(1)× U(1) AAVS 0 Y
495 23544 14 U(3)× U(2)× U(1)× U(1) AAVS 0
509 22156 17 U(3)× U(2)× U(1)× U(1) AAVS 0 Y
519 21468 13 U(3)× U(2)× U(1)× U(1) AAVA 0 Y
543 20176(*) 38 U(3)× U(2)× U(1)× U(1) VVVV 1/2 Y
617 16845 296 U(5)× O(1) AV 0 Y
671 14744(*) 29 U(3)× U(2)× U(1)× U(1) VVVV 1/2
761 12067 26 U(3)× U(2)× U(1) AAS 1/2 Y!
762 12067 26 U(3)× U(2)× U(1) AAS 0 Y!
1024 7466 7 U(3)× U(2)× U(2)× U(1) VAAV 1
1125 6432 87 U(3)× U(3)× U(3) VVV * Y
1201 5764(*) 20 U(3)× U(2)× U(1)× U(1) VVVV 1/2
1356 5856(*) 10 U(3)× U(2)× U(1)× U(1) VVVV 1/2 Y
1725 2864 14 U(3)× U(2)× U(1)× U(1) VVVV 1/2 Y
1886 2381 115 U(6)× Sp(2) AV 1/2 Y!
1887 2381 115 U(6)× Sp(2) AV 0 Y!
1888 2381 115 U(6)× Sp(2) AV 1/2 Y!
2624 1248 3 U(3)× U(2)× U(2)× U(3) VAAV 1
2880 1049 34 U(5)× U(1) AS 1/2 Y!
2881 1049 34 U(5)× U(1) AS 0 Y!
2807 1096(*) 8 U(3)× U(2)× U(1)× U(1) VVVV 1/2
2919 1024 2 U(3)× U(2)× U(2)× O(3) VAAV 1
4485 400(*) 2 U(3)× U(2)× U(1)× U(1) VVVV 1/2
4727 352 3 U(3)× U(2)× U(1)× U(1) VVVV 1/2
4825 332 20 U(4)× U(2)× U(2) VAS 1/2 Y!
4902 320(*) 1 U(3)× U(2)× U(1)× U(1) VVVV 1/2 Y
4996 304 30 U(3)× Sp(2)× U(1)× U(1) VVVV 1/2 Y
6993 128(**) 1 U(3)× U(2)× U(2)× U(1) VVVV 1/2
7053 124 4 U(3)× U(2)× U(2)× U(1) VASV 1/2 Y!
7241 116(**) 4 U(3)× U(2)× U(2)× U(1) VVVV 1/2
7280 114 3 U(3)× Sp(2)× U(1) AVS 1/2
7464 108 1 U(3)× Sp(2)× U(1) VVT 1/2
7905 96(*) 1 U(3)× U(2)× U(1)× U(1) VVVV 1/2
8747 68(**) 3 U(3)× U(2)× U(1)× U(1) VVVV 1/2
8773 68 4 U(3)× U(2)× U(1)× U(1) VVVV 1/2
11347 32(**) 1 U(3)× U(2)× U(1)× U(1) VVVV 1/2
Continued on next page
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Low energy predictions from string theory in four easy steps
2. Compute presently measurable low energy parameters Φ(continuous and discrete) of vacua with high accuracy.
# Generationsg
g
1
2
rk G
= computing map ~m 7→ Φ(~m).
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Low energy predictions from string theory in four easy steps
3. Find unique vacuum compatible with experiment.
= find ~m such that Φ(~m) = Φexp.
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Low energy predictions from string theory in four easy steps
4. Use this vacuum to predict everything we ever wanted to know.
m = ...
g = ...
G = ...*
*
*
= compute Φ̃everything we everwanted to know(~m).
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Is this in principle a tractable problem?
?
Computational complexity theory
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2027: string theory under full control!
Imagine we have systematic classification of all vacua, and that wecan compute for each vacuum every low energy quantity toarbitrarily high accuracy.
Now what?
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Model for matching observed data with microscopic data
ε
Λ0
E.g. cosmological constant in Bousso-Polchinski model:
Λ(N) = −Λ0 +∑ij
gijNiN j
with flux N ∈ ZK . Example question: ∃N : 0 ≤ Λ(N) < ε ?
Can be extended to more complicated models, other parameters, ...
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Computational complexity
I Bousso-Polchinski problem can be shown to be NP-complete.(Essentially due to high “nonlocality” of map N 7→ Λ(N).)
I ⇒ If you find polynomial time algorithm to solve this,everything we ever thought is difficult is actually easy, and youwin a $106 Clay prize.
I But more likely: no general polynomial time algorithm for BP(not even on quantum computer)
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Caveats
many