MODULI STABILISATION AND THE HOLOGRAPHIC SWAMPLAND Joseph Conlon - - PowerPoint PPT Presentation

MODULI STABILISATION AND THE HOLOGRAPHIC SWAMPLAND

Joseph Conlon

DIAS, Oct 2020 (based on JC, Quevedo 1811.06276, JC, Revello 2006.01021)

MODULI: WHAT?

• What is in ?
• Simple concept: massive scalar with

gravitationally suppressed couplings to ordinary matter such as

• Such moduli are well motivated from e.g. string

theory and extra-dimensional theories

L = L

GR + L SM + LBSM

LBSM

Φ M P F

µνF µν

Φ

MODULI: WHY?

• String theory is a theory of

dynamical extra dimensions

• In 4d theory, geometry of extra

dimensions (size and shape) parametrised by moduli - such as Kahler and complex structure moduli.

• Unstabilised, these lead to fifth forces, varying couplings or (fatal)

decompactification.

• Essential to develop moduli potentials that fix this geometry
• Stabilisation also provides a minimum in which to compute couplings

MODULI: WHY?

• String theory…..who cares?

MODULI: WHY?

• In an expanding universe
• As matter dominates over radiation, reheating is

dominated by the last fields to decay not the first

• The weaker the coupling, the longer the lifetime….
• Moduli potentials are everyone’s business

ρmatter ~ 1 a3 ρradiation ~ 1 a4

τ Φ ~ 8π M P

2

g2mΦ

3 ~ 10TeV

mΦ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3

10−3s

τ ~ 8π g2m

MODULI STABILISATION

• Much work on developing moduli potentials (LVS,

KKLT) and studying their dynamics with regards to Supersymmetry breaking Cosmology - late time de Sitter Cosmology - inflation Particle physics

MODULI STABILISATION

• String theory
• 10-dimensional supergravity with alpha’ corrections
• 4-dimensional supergravity of moduli and matter
• Integrate out heavy modes to get potential for

lightest moduli

• Find vacuum as minimum of effective potential

EFT EFT EFT EFT

LARGE VOLUME SCENARIO

Balasubramanian, Berglund, JC, Quevedo

• Perturbative corrections to K and non-perturbative

corrections to W

• Resulting scalar potential has minimum at exponentially large

values of the volume W = G3

∫

∧ Ω + A

i i

∑

e

−2πaiTi

K = −2ln V +ξ '

( )+ ln

Ω

∫

∧ Ω

( )− ln(S + S )

V = A τ se

−2asτ s

V − Bτ se

−asτ s

V 2 + C V 3

WHY LVS?

• In LVS, volume is exponentially large - can easily be
• This generates interesting hierarchies and ensures superb

parametric decoupling of heavy modes (KK modes, heavy moduli)

• Decoupling also has a clear geometric origin - large volume
• limit of LVS also leads to a unique effective

theory

〈V〉 ~ e

ξ/gs

V ~1050(2π ′ α )6

V → ∞

LVS HOLOGRAPHY

• LVS effective theory for volume modulus and

axion a

• Other terms are subleading in infinite volume limit by

Vpotential = V0e

−λΦ/ MP −

Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3/2

+ A ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

Lkinetic = 1 2 ∂µΦ∂µΦ + 3 4 e

− 8 3Φ ∂µ a∂µ a

O 1 lnV ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

λ = 27 / 2

( )

Φ

LVS HOLOGRAPHY

• LVS effective theory for volume modulus and

axion a

• Solve for minimum and expand about it to

determine masses and couplings

Vpotential = V0e

−λΦ/ MP −

Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3/2

+ A ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

Lkinetic = 1 2 ∂µΦ∂µΦ + 3 4 e

− 8 3Φ ∂µ a∂µ a

λ = 27 / 2

( )

Φ

HOLOGRAPHY

• CFT dimensions of dual operators:
• In infinite volume limit can classify modes as

heavy light interesting

Δ(Δ − 3) = mΦ

2RAdS 2

mΦ

2 ≫ RAdS −2 ,Δ → ∞

as V → ∞ mΦ

2 ≪ RAdS −2 ,Δ → 3

as V → ∞ mΦ

2 ~ RAdS −2 ,Δ → O(1−10)

asV → ∞

LVS MASS SPECTRUM

• In LVS we have
• Heavy: KK modes, complex structure moduli, all

Kahler moduli except overall volume

• Light: Graviton, overall volume axion
• Interesting: overall volume modulus

LVS HOLOGRAPHY

In minimal LVS, AdS effective theory has small number of fields which correspond to specific predictions for dual conformal dimensions

No Landscape!

(not true of KKLT)

LVS HOLOGRAPHY

• LVS is attractive as it offers a well-motivated

Generalised Free Field Theory

• Large volume limit gives a unique theory
• Two scalars with fixed and radiatively stable

anomalous dimensions

• All AdS interactions are also fixed and radiatively

stable

V → ∞

LVS AND THE SWAMPLAND

• LVS effective Lagrangian is
• This has the expected behaviour that

Vpotential = V0e

−λΦ/ MP −

Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3/2

+ A ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

Lkinetic = 1 2 ∂µΦ∂µΦ + 3 4 e

− 8 3Φ/ MP ∂µ a∂µ a

λ = 27 / 2

( )

fa / M P → 0 as V → ∞

LVS AND THE SWAMPLAND

• Now consider this small modification:
• This coupling is equivalent to axion decay constants

that diverge in the decompactification limit - must be in the swampland!

Vpotential = V0e

−λΦ/ MP −

Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3/2

+ A ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

Lkinetic = 1 2 ∂µΦ∂µΦ + 3 4 e

+ 8 3Φ/ MP ∂µ a∂µ a

λ = 27 / 2

( )

fa / M P → ∞ as V → ∞

HOLOGRAPHIC SWAMPLAND

• n-point self interactions of volume modulus
• Mixed interactions of volume modulus and axion
• The higher-point interaction define 3- and higher

point-correlators within a dual CFT

λ = 27 / 2

( )

Ln− pt = (−1)n−1λ n(n −1) −3 M P

2

RAdS

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 n! δ Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

1+O 1 lnV ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

L

Φnaa = − 8

3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

1 2n! δ Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

∂µ a∂µ a

HOLOGRAPHIC SWAMPLAND

• n-point self interactions of volume modulus
• Now modify interactions of volume modulus and axion
• This defines a perturbation to the Generalised Free Field CFT with

axion decay constants that diverge in the decompactification limit - must be in the swampland!

λ = 27 / 2

( )

Ln− pt = (−1)n−1λ n(n −1) −3 M P

2

RAdS

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 n! δ Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

1+O 1 lnV ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

L

Φnaa = + 8

3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

1 2n! δ Φ M P ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

n

∂µ a∂µ a

fa / M P → ∞ as V → ∞

HOLOGRAPHIC SWAMPLAND

• The problem:
• 1. Generalised Free Field + (some corrections)
• consistent theory
• 2. Generalised Free Field + (other corrections)
• swampland!

Where does the difference lie? Can one correlate any properties of the CFT with this change from the consistent theory to the swampland theory?

HOLOGRAPHIC SWAMPLAND

• 3-pt interactions in AdS theory relate to 3-pt

structure functions in CFT

• Signs of 3-pt functions are not determinate - can

change by field redefinitions

• Focus on 3-pt interactions only

HOLOGRAPHIC SWAMPLAND

• gives rise to CFT structure constants

HOLOGRAPHIC SWAMPLAND

• Anomalous dimensions are well-defined; can be

related to Mellin amplitude for 2 -> 2 scattering

HOLOGRAPHIC SWAMPLAND

HOLOGRAPHIC SWAMPLAND

Anomalous dimensions are equivalent to binding energies of 2-particle states in AdS In Mellin amplitude, ‘exchange’ t-channel diagrams provide dominant contribution at large l

HOLOGRAPHIC SWAMPLAND

LVS ‘just’ gives a negative anomalous dimension for the mixed volume-axion state ‘Correct’ signs in effective AdS equivalent to negative anomalies dimension for mixed

• perator

HOLOGRAPHIC SWAMPLAND

• In LVS context, right signs of 3-pt AdS couplings

are equivalent to negative anomalous dimensions for the mixed double-trace operator.

• A similar result holds for perturbative or KKLT

stabilisation (qualitatively different as involves a massive axion)

CONNECTION TO REFINED DISTANCE CONJECTURE

KK modes have to couple to light volume modulus in a way that their mass decreases with increasing volume This fixes the sign of the 3-pt function, again in a way that results in a negative anomalous dimension for the mixed double trace operator

CONCLUSIONS

• For many examples, negative CFT anomalous dimensions appear to

correspond to the correct signs in the AdS Lagrangian

• However:
• 1. Negativity of anomalous dimensions does not seem to hold for fibred

LVS with extra light fibre moduli

• 2. Axions couple with different signs to the different fibre moduli,

resulting in a mixture of signs

• Are earlier results just a feature of the volume modulus? In progress…..