Conjectures from Quantum Gravity - Exploring the Landscape inside - PowerPoint PPT Presentation

Conjectures from Quantum Gravity - Exploring the Landscape inside the Swampland - Florian Wolf Young Scientists Workshop at Castle Ringberg on July 19, 2017

Swampland vs. Landscape High energy, more dimensions, Quantum Gravity e.g. String Theory x Consistent 4 dim Landscape low energy effective theory Swampland [Vafa ’04] 2

Swampland vs. Landscape High energy, more dimensions, Quantum Gravity e.g. String Theory x Consistent 4 dim Landscape low energy effective theory Swampland [Vafa ’04] What should 4 dim EFT look like if and only if it arises from Quantum Gravity? There are (so far) two conjectures deciding between landscape and swampland. 2

Application to Stringy Large-Field Inflation Inflaton = axionic modulus from String Theoy Periodic potential Polynomial potential Trans-planckian axion decay Trans-planckian field movement: f > 1 M Pl constant: ∆ φ > 1 M Pl V ( θ ) θ What are axions? φ → φ + 2 π f Scalars equipped with discrete shift symmetry Some moduli of String Theory are axions 3

Application to Stringy Large-Field Inflation Inflaton = axionic modulus from String Theoy Periodic potential Polynomial potential Trans-planckian axion decay Trans-planckian field movement: f > 1 M Pl constant: ∆ φ > 1 M Pl V ( θ ) θ Constraints from Constraints from Weak Gravity Conjecture Swampland Conjecture [Arkani-Hamed, Motl, Nicolis, Vafa, …many more] [Vafa, Ooguri, Palti, Baume, Kläwer, Blumenhagen, Valenzuela, FW] What are axions? φ → φ + 2 π f Scalars equipped with discrete shift symmetry Some moduli of String Theory are axions 3

Outline 1. Introduction 2. Weak Gravity Conjecture • Electric and Magnetic Versions • Application to Periodic Inflation 3. Swampland Conjecture • Extension to Axions via Backreaction • Critical Distance and Polynomial Inflation 4. Conclusion 4

The Weak Gravity Conjecture (WGC) A simple observation of our world (and all consistent string compactifications): Gravity is the weakest force Promote to general principle 5

The Weak Gravity Conjecture (WGC) A simple observation of our world (and all consistent string compactifications): Gravity is the weakest force Promote to general principle Consider 4 dim theory with gravity and U(1) gauge field with coupling : g el There must exist a light charged particle Q with Electric WGC: [Arkani-Hamed, Motl, Nicolis, Vafa ’06] m el ≤ g el M Pl Q Q Gauge repulsion Gravity Gravity Gauge repulsion 5

Magnetic Weak Gravity Conjecture WGC formula should also hold for magnetic monopoles. 6

Magnetic Weak Gravity Conjecture WGC formula should also hold for magnetic monopoles. What are magnetic monopoles? Motivated by electric-magnetic symmetry of Maxwell’s Eq., Dirac studied particles with net magnetic charge g mag g el · g mag ∈ Z Dirac quantisation condition: 7

Magnetic Weak Gravity Conjecture WGC formula should also hold for magnetic monopoles. m mag ≤ g mag M Pl From Dirac’s quantisation condition Λ EFT has cutoff g mag ∼ 1 m mag ∼ g 2 mag Λ g el Λ ≤ g el M Pl Magnetic WGC: For small gauge coupling EFT breaks down at low scale! Unexpected from 4 dim EFT point of view 8

WGC for Axions and Inflation Generalising WGC to p-form gauge fields in arbitrary dimensions leads to axion version m ax ≤ g ax M Pl f Axion coupled to instanton with Axion decay constant S inst g ax ∼ 1 action m ax ∼ S inst f f · S inst ≤ M Pl Axionic WGC: 9

WGC for Axions and Inflation Generalising WGC to p-form gauge fields in arbitrary dimensions leads to axion version m ax ≤ g ax M Pl f Axion coupled to instanton with Axion decay constant S inst g ax ∼ 1 action m ax ∼ S inst f f · S inst ≤ M Pl Axionic WGC: Consequence for inflation: ✓ θ ◆ instanton generates dangerous terms V ( θ ) ∼ e − S inst cos + . . . f in inflaton potential: S inst > 1 Flat potential for slow-roll inflation requires: WGC implies: no trans-planckian axion decay constants 9

WGC for Axions and Inflation Generalising WGC to p-form gauge fields in arbitrary dimensions leads to axion version m ax ≤ g ax M Pl f Axion coupled to instanton with Axion decay constant S inst g ax ∼ 1 action m ax ∼ S inst f f · S inst ≤ M Pl Axionic WGC: Consequence for inflation: ✓ θ ◆ instanton generates dangerous terms V ( θ ) ∼ e − S inst cos + . . . f in inflaton potential: No periodic S inst > 1 Flat potential for slow-roll inflation requires: inflation WGC implies: no trans-planckian axion decay constants 9

Outline 1. Introduction 2. Weak Gravity Conjecture • Electric and Magnetic Versions • Application to Periodic Inflation 3. Swampland Conjecture • Extension to Axions via Backreaction • Critical Distance and Polynomial Inflation 4. Conclusion 10

The Swampland Conjecture Moduli = free parameter emerging during compactification d ( p 0 , p ) → ∞ For an infinite p tower of massive states becomes Non-axionic moduli space exponentially light: [Ooguri, Vafa ‘04] d ( p 0 , p ) M ∼ M 0 e − α d ( p 0 ,p ) p 0 for theories in the landscape Parameter a priori undetermined α 11

The Swampland Conjecture Moduli = free parameter emerging during compactification d ( p 0 , p ) → ∞ For an infinite p tower of massive states becomes Non-axionic moduli space exponentially light: [Ooguri, Vafa ‘04] d ( p 0 , p ) M ∼ M 0 e − α d ( p 0 ,p ) p 0 for theories in the landscape Parameter a priori undetermined α Consequence: d ( p 0 , p ) > 1 EFT invalid if traversing distance in non-axionic α moduli space! 11

Extension to axions via Backreaction I ( s, θ ) Generate a potential for moduli by turning on background fluxes. θ Move one axionic modulus - called inflaton - from minimum. Backreaction: other moduli vev adjust according to inflaton movement. s Min 12

Extension to axions via Backreaction I ( s, θ ) Generate a potential for moduli by turning on background fluxes. θ Move one axionic modulus - called inflaton - from minimum. Backreaction: other moduli vev adjust according to inflaton movement. s Min s Min ( θ ) = s Min + λ θ 12

Extension to axions via Backreaction I ( s, θ ) Generate a potential for moduli by turning on background fluxes. θ Move one axionic modulus - called inflaton - from minimum. Backreaction: other moduli vev adjust according to inflaton movement. s Min Strong s Min ( θ ) ≈ λ θ s Min ( θ ) = s Min + λ θ backreaction 12

Extension to axions via Backreaction I ( s, θ ) Generate a potential for moduli by turning on background fluxes. θ Move one axionic modulus - called inflaton - from minimum. Backreaction: other moduli vev adjust according to inflaton movement. s Min Strong s Min ( θ ) ≈ λ θ s Min ( θ ) = s Min + λ θ backreaction Kinetic term for axion derived from String Theory: 1 L θ ( ∂θ ) 2 kin ∼ s 2 Min 12

Extension to axions via Backreaction I ( s, θ ) Generate a potential for moduli by turning on background fluxes. θ Move one axionic modulus - called inflaton - from minimum. Backreaction: other moduli vev adjust according to inflaton movement. s Min Strong s Min ( θ ) ≈ λ θ s Min ( θ ) = s Min + λ θ backreaction Kinetic term for axion derived from String Theory: 1 1 Strong L θ ( λθ ) 2 ( ∂θ ) 2 L θ ( ∂θ ) 2 kin ≈ kin ∼ s 2 backreaction Min 12

Extension to axions via Backreaction II kin ∼ 1 L θ 2 ( ∂ Θ ) 2 Canonical normalisation: implies Θ ∼ exp( λθ ) 13

Extension to axions via Backreaction II kin ∼ 1 L θ 2 ( ∂ Θ ) 2 Canonical normalisation: implies Θ ∼ exp( λθ ) Consequence: Some heavy modes (e.g. KK- or string modes) which have been integrated out in EFT, become light: 1 1 Strong θ ∼ e − λ Θ M heavy ∼ s Min ( θ ) backreaction Θ c ∼ 1 EFT invalid above critical distance λ Swampland Conjecture for axions [Palti, Baume/Kläwer ’16] 13

What is the Critical Field Range? - An Illustrative Model - Model on isotropic 6-torus with one D7-brane position modulus. [Blumenhagen, Valenzuela, FW] Superpotential: W = f 0 + 3 f 2 U 2 − h S U − q T U − µ Φ 2 complex structure axio-dilaton Kähler open string modulus with quantised fluxes f , f 2 , h, q, µ 14

What is the Critical Field Range? - An Illustrative Model - Model on isotropic 6-torus with one D7-brane position modulus. [Blumenhagen, Valenzuela, FW] Superpotential: W = f 0 + 3 f 2 U 2 − h S U − q T U − µ Φ 2 complex structure axio-dilaton Kähler open string modulus f , f 2 , h, q, µ with quantised fluxes Kähler potential: ⇥ ( S + S )( U + U ) − 1 2 ( Φ + Φ ) 2 ⇤ K = − 3 log( T + T ) − 2 log( U + U ) − log 14

What is the Critical Field Range? - An Illustrative Model - Model on isotropic 6-torus with one D7-brane position modulus. [Blumenhagen, Valenzuela, FW] Superpotential: W = f 0 + 3 f 2 U 2 − h S U − q T U − µ Φ 2 complex structure axio-dilaton Kähler open string modulus f , f 2 , h, q, µ with quantised fluxes Kähler potential: ⇥ ( S + S )( U + U ) − 1 2 ( Φ + Φ ) 2 ⇤ K = − 3 log( T + T ) − 2 log( U + U ) − log Compute the F-term scalar potential for moduli: V F = M 4 e K ⇣ � 2 ⌘ K IJ D I WD J W − 3 � � Pl � W 4 π 14