Distribution of Vacua in Calabi-Yau Compactification Yuji - - PowerPoint PPT Presentation
Distribution of Vacua in Calabi-Yau Compactification Yuji Tachikawa (Particle Theory Group, Univ. of Tokyo, Hongo) based on JHEP 01 (2006) 100 [hep-th/0510061] by Tohru Eguchi and YT March, 2006 @ UTAP , Hongo 0/68 CONTENTS 1. On the
(Particle Theory Group, Univ. of Tokyo, Hongo) based on JHEP 01 (2006) 100 [hep-th/0510061] by Tohru Eguchi and YT
0/68
CONTENTS
1. On the Landscape & the Swampland
⋄
2. Flux Compactification
⋄
3. Statistics of Vacua
⋄
4. Monodromy and Vacuum Density
⋄
5. Summary & Comments
1/68
⋄
Quantization of gravity
⇐ spectral index of primordial fluctuation ⋄
Candidates (generally covariant + quantum mechanical):
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String / M theory
⋄
Not originally meant to quantize gravity
⋄
Worldlines ⇒ Worldsheets
⋄
Consistency ⇒ 10 D + graviton
⋄
Many higher-dim’l solitons, branes, which support gauge fields
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Compactification
⋄
⋄
Many consistency conditions.
⋄
Semi-realistic models:
which is a triumph for string theory.
⋄
Presence of Moduli.
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Status
⋄
No experimental tests.
⋄
Rich as a theoretical model
(ADHM, Seiberg-Witten, Montonen-Olive duality etc.)
⋄
Unified most of the research on QFT & SUGRA practitioners
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Moduli Fields
⋄
⋄
How light ? ⇒ massless or SUSY br. or Hubble
⋄
Corresponding to the ‘moduli’ of the compactification manifold
⋄
⋄
the shape & size of the internal manifold.
⋄
Shape & size determines the Yukawa/gauge couplings.
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Moduli Problem
⋄
Massless scalar ⇒ 5th force
⋄
Susy breaking will make them massive ∼ Msb,
etc.
⋄
Need to make it much heavier !
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Moduli Fixing in String theory
⋄
Vexing problems for a long time
⇐ Consistency forbids introduction of potentials by hand ⋄
day.
⋄
Roughly speaking,
⇒ Tend to spread
⇒ Tend to shrink ⇒ Shape & Size fixed.
8/68
⋄
# of choices of flux are HUGE !!
⇒ 100100 of choices ⋄
Flux given ⇒ Moduli fixed
⇒ Shape & size fixed ⇒ Yukawa & gauge coupling ⋄
⋄
Huge landscape of 4d vacua.
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Really?
⋄
Opinion varies:
⇑
⇓
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Swampland [Vafa]
⋄
we’d like to argue without the long detour into 10d string, Calabi-Yau, fluxes and all that messy stuffs.
⋄
Anomaly cancellation.
⇒ Certain gauge groups & matter contents are not allowed. ⋄
Upperbound on the rank of gauge groups
⋄
Gravity should be weaker than gauge coupling [Arkani-Hamed-Motl-Nicolis-Vafa, hep-th/0601001]
⋄
Positivity of certain dimension > 4 operators ⇐ Causality. [Adams-Arkani-Hamed-Dubovsky-Nicolis-Rattazzi , hep-th/0602178]
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CONTENTS
On the Landscape & the Swampland
2. Flux Compactification
⋄
3. Statistics of Vacua
⋄
4. Monodromy and Vacuum Density
⋄
5. Summary & Comments
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d = 4, N = 1 Supergravity ⋄ {Qα, Qβ} = γµ
αβPµ
µ, λa α)
α, φi)
⋄ Pµ gauged ⇒ Qα gauged ⋄ φi are complex scalars, Gi¯
and V restricted in
(φ, ¯
φ)∂µφi∂µ ¯ φ¯
+ V (φ, ¯
φ)
⋄ K(φ, ¯ φ): K¨
Gi¯
(φ, ¯
φ) = ∂i ¯ ∂¯
K(φ, ¯
φ), V (φ, ¯ φ) = eK Gi¯
DiW (φ) ¯
D¯
¯
W ( ¯ φ) − 3|W (φ)|2 DiW (φ) = (∂i + (∂iK))W ⋄
K¨ ahler transformation:
K(φ, ¯ φ) → K(φ, ¯ φ) + f(φ) + ¯ f( ¯ φ) W (φ) → e−f(φ)W (φ) DiW (φ) → e−f(φ)DiW (φ)
leaves Gi¯
and V (φ, ¯
φ) invariant.
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10d IIB supergravity
e−φ, C, gµν, HNSNS
[µνρ] = ∂[µBNSNS νρ]
,
HRR
[µνρ] = ∂[µBRR νρ],
F[µνρστ] = ∂[µCνρστ] with constraint F[µνρστ] = ǫµνρσταβγδǫF [αβγδǫ],
+fermions An important coupling:
(3)
∧ HRR
(3)
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Branes
⋄
point-like objects couple to Aµ via
dxµAµ ⋄
dxµ0 · · · dxµpC[µ0···µp]
⇐
D(-1) brane = D-instanton
⇐
F1 brane = string
⇐
D1 brane = D-string
⇐
D3 brane
⋄
(3)
∧ HRR
(3) ⇒ HNSNS ∧ HRR has D3-brane charge
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Calabi-Yau compactification
⋄
10=4+6
⋄
6-dimensional CY = the holonomy SU(3) ⊂ SO(6)
⇒ CY : complex mfd x1, x2, x3, x4, x5, x6 → z1, z2, z3, ¯ z¯
1, ¯
z¯
2, ¯
z¯
3,
with K¨ ahler form ω, everywhere nonzero (3, 0) form Ω
⋄
6d spinor 4 = 3 ⊕ 1 under SU(3)
⇒ 1/4 of SUSY remain ⇒ Type IIB/CY : N = 2 ⋄
No gauge fields ⇒ put D-branes
⇒ breaks SUSY to N = 1
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Moduli in CY compactification
⋄
CYs come in various topological types:
A0, A1, . . . , Ah12 and B0, B1, . . . , Bh12
so that Ai · Bj = δij and Ai · Aj = Bi · Bj = 0
⋄
CYs can be continuously deformed , parametrized by
ω ∧ ω : sizes of four-cycles for i = 1, . . . , h11
Ω : periods of three-cycles for i = 1, . . . , h12
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⋄
The metric of CY varies as ρi and zi :
⇒ 10d metric : ds2 = ηµνdxµdxν + gmn(ρi(xµ), zi(xµ))dxmdxn ⋄
Plug this into S =
S =
+
(4)Gi¯ ∂µρi∂ν ¯
ρ¯
+
(4)G′ i¯ ∂µzi∂ν ¯
z¯
⋄ ρi combines with
C(4) to become a complex scalar ρi
complexified = i
ω ∧ ω +
C(4)
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⋄ h11 + h12 massless complex scalars in total
⋄
Axio-dilaton τ = ie−φ + C(0) is also a modulus.
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Superpotentials for Moduli
⋄
Just compactifying on CY leads to W = 0 ⇒ V = 0.
⋄
Masses to all moduli
⇒ We need W depending all variables τ, ρi, zi.
[Kachru-Kallosh-Linde-Trivedi hep-th/0301240]
⋄
Let’s see each in detail.
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Flux superpotential
⋄
Type IIB has 2-form potentials BNSNS and BRR with 3-form field strengths HNSNS and HRR
⋄
Quantized fluxes through three-cycles
⋄
They give rise to
W =
Ω ∧ (HRR + τHNSNS) =
h12
Ω
(HRR + τHNSNS) −
Ω
(HRR + τHNSNS)
h12
i
+ τNNSNS
i
) − ∂F ∂zi (MRR
i
+ τMNSNS
i
)
Comments
W =
h12
i
+ τNNSNS
i
) − ∂F ∂zi (MRR
i
+ τMNSNS
i
)
This depends on string coupling and shape, not on the size .
⋄ Ni and Mi are the number of fluxes, hence integers ⋄
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⋄
This form for W : obtainable by a standard KK reduction;
⋄
⇒ a BPS domain wall in 4d point of view. ⇒ The tension should be
Ω
⇒ W !
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Constraint on Ni and Mi
⋄
A term
⋄
Of course there is a coupling
C(4). ⋄
Another coupling −
C(4) to Orientifold planes. ⇒ EOM for C(4) leads #O3 = #D3 +
= #D3 +
h12
i MNSNS i
− MRR
i NNSNS i
#O3 is fixed by the geometry of CY.
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Instanton corrections
⋄
Superpotentials for the size moduli ρi : How?
⋄
⇒ N = 1 U(N) gauge theories with coupling constant ρi ⇒ Superpotential ∼ e−iρi/N
associated with gaugino condensation.
⋄
⇒ Contributes ∝ e−iρi to the superpotential
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⋄ ∃ CYs with sufficiently generic instanton corrections [Denef-Douglas]. ⇓
⋄
Caveats:
ρ) might have bigger effects. [Conlon-Quevedo]
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⋄
Further caveats:
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CONTENTS
On the Landscape & the Swampland
Flux Compactification
3. Statistics of Vacua
⋄
4. Monodromy and Vacuum Density
⋄
5. Summary & Comments
29/68
⋄
We used fluxes HRR and HNSNS.
⋄
In a typical CY, there’re 100∼200 3-cycles to put fluxes;
⋄
LHS of the tadpole constraint
#O3 = #D3 +
is of order 1000∼5000.
⋄
SUSY requires #D3 ≥ 0 and the quadratic form positive definite
⇒
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⋄
Gauge group & matter contents : ⇐ topology of the CY
⋄
Coupling constants ⇐ the moduli ⇐ Flux
⋄
Once you construct the SM (+ susy + inflatons etc.), there’ll be plethora of vacua with slightly differing Yukawas!
31/68
⋄
Need the distribution of Yukawas / Cosmological constants
⋄
which are determined by the moduli
⇒ We need the distribution of the moduli ! ⋄
Fixed moduli depends on the flux ...
⇒ Need the distribution of HRR and HNSNS.
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We don’t know yet.
⋄
Fluxes change when we cross domain walls.
⇒ Flux distribution is tied to the dynamics of domain walls
in the extremely early universe before inflation!
⋄
So we can’t study realistic distribution of flux. Period.
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As a zeroth approximation,
⋄
We try a gaussian ensemble of the fluxes HRR and HNSNS:
Ni =
(HRR + τHNSNS), Mi =
(HRR + τHNSNS). ⋄
Under a large fluctuation, we have monodromies acting on (Ni, Mi):
Mi
B C D Ni Mi
(NiMi′ − Ni′Mi) ⋄
Assume the ensemble to be monodromy invariant.
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⋄
Distribution of W (z) = Nizi − Mi
∂F ∂zi ⇒ W (z)W (w)∗ ∝
∂F ∂wi ∗ − wi∗ ∂F ∂zi
W (z)W (w) = 0 W (z)∗W (w)∗ = 0
35/68
⋄ W (z)W (w)∗ ∝ e−K(z,w∗) is very natural ,
because it transforms covariantly under the K¨ ahler transform:
K(z, z∗) → K + f(z) + f∗(z∗), W (z) → e−f(z)W (z) ⋄
We can study the behavior of N = 1
with W (z)W (w)∗ ∝ e−K(z,w∗).
⋄
Huge literature on systems with random potential (not superpotential) in condensed matter physics. We should utilize them...
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Distribution of Vacua
⋄
Supersymmetric Vacua are defined by DiW = 0.
⇒ Expected number of vacua at zi is given by ρ(z, ¯ z) = δ(DiW (z))δ( ¯ D¯
ıW (¯
z)∗)
∂iDjW ∂iD¯
W ∗
∂¯
ıDjW
∂¯
ıD¯ jW ∗
Determinant needed to count each vacua with weight +1.
⋄
Absolute value makes evaluation harder; instead consider
˜ ρ(z, ¯ z) = δ(DiW (z))δ( ¯ D¯
ıW (¯
z)∗) det ∂iDjW ∂iD¯
W ∗
∂¯
ıDjW
∂¯
ıD¯ jW ∗
This counts vacua with signs ±1.
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⋄ ˜ ρ can be calculated using Wick’s theorem. ⋄
The result is,
˜ ρ(z)
dzi ∧ d¯ z¯
ı ∝ det 1
2π(Rij + δijω)
where
Rij = Ri
ik¯ l dzk ∧ d¯
z¯
l,
ω = i 2gi¯
dzi ∧ d¯
z¯
is the curvature and the K¨ ahler form of the moduli space .
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A mathematical comment
⋄
Let M compact n dim’l K¨ ahler and nonsingular,
⋄ E a n dim’l vector bundle on M. ⇒ A generic section of E have
e(E) zeros,
when counted with signs, where e(E) is the Euler class.
⋄ e(E) = det RE via the Chern-Weil homomorphism. ⋄ DiW is a section of T M ⊗ H⇒
det RT M⊗H =
det(RT M + RH) =
det(RT M + ω) ⋄
In supergravity M is noncompact and singular !
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Physical Comments
⋄
Suppose there’re no curvature : R = 0. ⇒ ˜
ρ ∝ det ω ⇒ the vacua distribute uniformly following the volume. ⋄
Vacua tends to cluster around where the curvature R is large.
⋄
Recall we’re discussing the curvature of the moduli space.
⋄
Curvature of the moduli is large ⇔ the curvature of the CY is large.
⇒ Strongly curved extra dimension is favored .
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Examples
⋄
To visualize ˜
ρ, ⋄
We need to calculate gi¯
and Rij :
gi¯
= ∂i ¯
∂¯
K,
Ri
jk¯ l = ∂¯ lgj ¯ m∂kg ¯ mi
⇒ Consult the mirror symmetry literature, ⇒ Plug into the formula for ˜ ρ, ⇒ Now you have a distribution of vacua !
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Near Conifold Singularity[Denef-Douglas, Giryavets-Kachru-Tripathy]
⋄
where a 3-cycle collapses. Call it A1.
⋄
Let φ ≡ X1 ⇒ F1 ∼ φ log φ:
gφφ∗ ∼ log(|φ|2), Rφφ∗ ∼ 1 |φ|2(log |φ|)2 ≫ gφφ∗
200 400 600 800 1000 0.002 0.004 0.006 0.008 0.01 xR or xg x
42/68
Two param. example [Eguchi-Y.T., unpublished]
⋄
Took two-modulus CY: degree 8 hypersurface in WCP4
1,1,2,2,2 with
1 8x8
1 + 1
8x8
2 + 1
8x4
3 + 1
8x4
4 + 1
8x4
5 − ψ0x1x2x3x4x5 − 1
4ψs(x1x2)4 = 0 ⋄
ples from supergravity.
⋄
Denote ǫ = 1/(2ψs) and u = ψ + ψ4
ǫ1/2 : Dynamical Scale of SYM measured in Planck units; u : Seiberg-Witten’s u.
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ǫ = 0.001, u : finite ⋄
Just two conifold singularities at u = ±1.
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u = 5, vary ǫ ⋄ det(R + ω) ∼ 1 |ǫ|1(log |ǫ|)3 if 1/ǫ ≫ u ≫ 1
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CONTENTS
On the Landscape & the Swampland
Flux Compactification
Statistics of Vacua
4. Monodromy and Vacuum Density
⋄
5. Summary & Comments
46/68
Singularity in Moduli
⋄
Related to the singularity in CY
⋄
Example: Conifold Singularity
x2 + y2 + z2 + w2 = ǫ
where x, y, z, w ∈ C
⋄
Easier Example: A1 Singularity
x2 + y2 + z2 = ǫ
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⋄
Much easier example:
x2 + y2 = ǫ
Suppose ǫ ∈ R>0 ⇒
Circle;
Re x2 − Im y2 = ǫ ⇒
Hyperbola
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x2 + y2 + z2 = ǫ − → x2 + y2 = ǫ − z2
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x2 + y2 + z2 + w2 = ǫ − → x2 + y2 + w2 = ǫ − z2
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A-cycle B-cycle
Monodromy
51/68
z =
Ω, A → A; Fz =
Ω, B → A + B. ⇓ z → z, Fz → z + Fz.
As z ∼ ǫ + O(ǫ2),
z ∼ ǫ, Fz ∼ ǫ 2πi log ǫ.
52/68
Special K¨ ahler geometry
⋄
Existence of special coordinates X0, · · · , Xn and the prepotential F (X) so that
e−K = ¯ XIFI − ¯ FIXI,
where
FI = ∂F ∂XI . ⋄
For the complex structure moduli of Calabi-Yau,
XI =
Ω, Fi =
Ω.
where AI · AJ = BI · BJ = 0, AI · BJ = δIJ
⋄
Parameters are zi = Xi/X0, (i = 1, 2, . . . , n).
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Vacuum counting in Calabi-Yau moduli
⋄
Singularity in CY
⇒ Singularity in the moduli ⇒ monodromy in X and F ⇒ the divergent behavior of X and F from holomorphy ⇒ e−K = ¯ XIFI − ¯ FIXI ⇒ gi¯
= ∂i ¯
∂¯
K
⇒ Curvature.
54/68
⋄
For K¨ ahler manifolds with gi¯
= ∂i ¯
∂¯
K,
Ri¯
k¯ l = gi ¯ m∂kg ¯ mn ¯
∂¯
lgn¯
⋄
For Special K¨ ahler manifolds, Strominger’s formula states
Ri¯
k¯ l = −e2KFikm ¯
F¯
¯ l¯ n g¯ nm + gi¯ gk¯ l + gi¯ lgk¯
where
Fijk = XI∂i∂j∂kFI − FI∂i∂j∂kXI
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Comments
⋄
Special K¨ ahler geometry emerged independently from :
⋄
String theory provides the reason of this ‘coincidence’.
⋄
Special K¨ ahler gemetry was crucial to
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Conifold Singularity
⋄
As ǫ goes round 0, X1 → X1 and F1 → F1 + X1 ⇒
X1 ∼ ǫ
and
F1 ∼ ǫ 2πi log ǫ ⇒ K = ¯ ǫǫ log |ǫ| ⇒ gǫ¯
ǫ = ∂ ¯
∂K = log |ǫ| ⇒ Rǫ¯
ǫ = ∂ǫ g¯ .. ¯
∂¯
ǫ g.¯ . =
1 |ǫ|2(log |ǫ|)2 ⇒
dǫd¯ ǫ |ǫ2|(log |ǫ|)2 < ∞ ⋄
Density det(R + g) strongly peaked near ǫ ∼ 0,
⋄
Integral is finite.
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What about other singularities ?
⋄
Many other kinds of singularity in Calabi-Yau :
etc.
⋄
Is the enhancement always finite ?
58/68
Our result:
⋄
Codimension d singularity
⇐ Need to tune d complex parameters to get to the singularity
59/68
Sketch of the derivation
⋄
Possible Monodromy : constrained by a mathematical theorem
⇒ X and F ⇒ K¨
ahler form ⇒ Metric ⇒ Curvature
⋄
Need upper bounds for each term in curvature
⇐ Easy
i ⇐ lower bound for gi¯
⇐ Polarization of the mixed Hodge structure of the singularity
60/68
A bit more detail
⋄ (Xi, Fi) → M(Xi, Fi) for ǫ → e2πiǫ
⋄
Take k s.t. eigenvalues of Mk = 1, and change the parameter a = ǫk.
⋄ N = Mk − 1 satisfies N4 = 0 ⇒
Fi
N 2πi log a
Xi(0) Fi(0)
Xi(1) Fi(1)
Xi(2) Fi(2)
⋄
Take p s.t. Np(Xi(0), Fi(0))T = 0 but Np+1(Xi(0), Fi(0))T = 0 .
⇒ (Xi, Fi) (log a)p ⋄
many e−K = ¯
XiFi − ¯ FiXi in the denominator in the expansion ⇒ Needs lower bound for ¯ XiFi − ¯ FiXi ⋄
Leading behavior
¯ XiFi − ¯ FiXi ∼ ( ¯ Xi(0)NpFi(0) − ¯ Fi(0)NpXi(0))(log a)p + · · · ⋄
A deep mathematical fact ensures ( ¯
XiNpFi − ¯ FiNpXi)(0) = 0 ⇒ eK = ( ¯ XiFi − ¯ FiXi)−1 (log a)−p ⇒ · · · ⇒ Integral converges !
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⋄
Explicitly studied two cases:
⇐ Electron and Monopole become simultaneously massless
⇐ Yang-Mills theory decouples from gravity
63/68
Argyres-Douglas singularity
⋄
Local form x2 + y2 + w2 = z3 − 3az − 2b with moduli a, b
64/68
⋄
Roots of z3 − 3az − 2b = 0 determines the singularity
2 3
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⋄
What happens near a ∼ b ∼ 0 ?
s=t t=0 s=0 DAD D2 Dc D3 a s D2 Dc a b Dc
b=as a=st t=sα
s=0 t=0 s=t D2 D3 Dc
⋄
Nothing in particular !
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CONTENTS
On the Landscape & the Swampland
Flux Compactification
Statistics of Vacua
Monodromy and Vacuum Density
5. Summary & Comments
67/68
68/68