MODULI STABILISATION AND THE HOLOGRAPHIC SWAMPLAND Joseph Conlon DIAS, Oct 2020 (based on JC, Quevedo 1811.06276, JC, Revello 2006.01021)
MODULI: WHAT? L = L GR + L SM + L BSM • What is in ? L BSM Φ • Simple concept: massive scalar with gravitationally suppressed couplings to ordinary Φ matter such as µ ν F µ ν F M P • Such moduli are well motivated from e.g. string theory and extra-dimensional theories
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 ρ matter ~ 1 ρ radiation ~ 1 a 3 a 4 • 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…. 3 τ ~ 8 π ⎛ ⎞ τ Φ ~ 8 π M P 2 3 ~ 10TeV 10 − 3 s ⎜ ⎟ g 2 m ⎝ ⎠ g 2 m Φ m Φ • Moduli potentials are everyone’s business
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 EFT • 10-dimensional supergravity with alpha’ corrections EFT • 4-dimensional supergravity of moduli and matter EFT • Integrate out heavy modes to get potential for lightest moduli EFT • Find vacuum as minimum of effective potential
LARGE VOLUME SCENARIO Balasubramanian, Berglund, JC, Quevedo • Perturbative corrections to K and non-perturbative corrections to W ∑ ∫ − 2 π a i T i W = ∧ Ω + G 3 A e i i ( ) − ln( S + S ) ( ) + ln ∫ K = − 2ln V + ξ ' Ω ∧ Ω • Resulting scalar potential has minimum at exponentially large values of the volume − 2 a s τ s V = A τ s e − a s τ s − B τ s e + C V 2 V 3 V
WHY LVS? • In LVS, volume is exponentially large - can easily be V ~10 50 (2 π α ) 6 ′ • 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 ξ / g s 〈 V 〉 ~ e • limit of LVS also leads to a unique effective V → ∞ theory
LVS HOLOGRAPHY Φ • LVS effective theory for volume modulus and axion a ( ) λ = 27 / 2 ⎛ ⎞ 3/2 ⎛ ⎞ Φ − λ Φ / M P − V potential = V 0 e + A ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ M P ⎠ ⎝ ⎠ 3 Φ ∂ µ a ∂ µ a 8 L kinetic = 1 2 ∂ µ Φ∂ µ Φ + 3 − 4 e • Other terms are subleading in infinite volume limit by ⎛ ⎞ 1 O ⎜ ⎟ ⎝ ⎠ ln V
LVS HOLOGRAPHY • LVS effective theory for volume modulus and Φ axion a ⎛ ⎞ 3/2 ⎛ ⎞ Φ − λ Φ / M P − ( ) V potential = V 0 e + A ⎜ ⎟ ⎜ ⎟ λ = 27 / 2 ⎜ ⎟ ⎝ M P ⎠ ⎝ ⎠ 3 Φ ∂ µ a ∂ µ a 8 L kinetic = 1 2 ∂ µ Φ∂ µ Φ + 3 − 4 e • Solve for minimum and expand about it to determine masses and couplings
HOLOGRAPHY • CFT dimensions of dual operators: Δ ( Δ − 3) = m Φ 2 R AdS 2 • In infinite volume limit can classify modes as 2 ≫ R AdS − 2 , Δ → ∞ V → ∞ heavy m Φ as 2 ≪ R AdS − 2 , Δ → 3 light V → ∞ m Φ as interesting 2 ~ R AdS − 2 , Δ → O (1 − 10) as V → ∞ m Φ
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 V → ∞ • Two scalars with fixed and radiatively stable anomalous dimensions • All AdS interactions are also fixed and radiatively stable
LVS AND THE SWAMPLAND • LVS effective Lagrangian is ⎛ ⎞ 3/2 ⎛ ⎞ ( ) Φ − λ Φ / M P − λ = 27 / 2 V potential = V 0 e + A ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ M P ⎠ ⎝ ⎠ 8 3 Φ / M P ∂ µ a ∂ µ a L kinetic = 1 2 ∂ µ Φ∂ µ Φ + 3 − 4 e • This has the expected behaviour that f a / M P → 0 V → ∞ as
LVS AND THE SWAMPLAND • Now consider this small modification: ⎛ ⎞ 3/2 ⎛ ⎞ ( ) Φ − λ Φ / M P − λ = 27 / 2 V potential = V 0 e + A ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ M P ⎠ ⎝ ⎠ 8 3 Φ / M P ∂ µ a ∂ µ a L kinetic = 1 2 ∂ µ Φ∂ µ Φ + 3 + 4 e • This coupling is equivalent to axion decay constants that diverge in the decompactification limit - must be in the swampland! f a / M P → ∞ V → ∞ as
HOLOGRAPHIC SWAMPLAND • n-point self interactions of volume modulus n ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 δ Φ L n − pt = ( − 1) n − 1 λ n ( n − 1) − 3 M P 1 1 1 + O ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 ⎝ ⎠ n ! ⎝ M P ⎠ ln V ⎝ R AdS ⎠ ( ) λ = 27 / 2 • Mixed interactions of volume modulus and axion n n ⎛ ⎞ ⎛ ⎞ δ Φ Φ n aa = − 8 1 ∂ µ a ∂ µ a L ⎜ ⎟ ⎜ ⎟ 3 2 n ! ⎝ M P ⎠ ⎝ ⎠ • The higher-point interaction define 3- and higher point-correlators within a dual CFT
HOLOGRAPHIC SWAMPLAND • n-point self interactions of volume modulus n ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 δ Φ L n − pt = ( − 1) n − 1 λ n ( n − 1) − 3 M P 1 1 1 + O ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 ⎝ ⎠ n ! ⎝ M P ⎠ ln V ⎝ R AdS ⎠ ( ) λ = • Now modify interactions of volume modulus and axion 27 / 2 n n ⎛ ⎞ ⎛ ⎞ δ Φ Φ n aa = + 8 1 ∂ µ a ∂ µ a L ⎜ ⎟ ⎜ ⎟ 3 2 n ! ⎝ M P ⎠ ⎝ ⎠ • 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! f a / M P → ∞ V → ∞ as
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 operator
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…..
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