.. Gauge/Gravity Duality: An introduction Johanna Erdmenger Max Planck Institut für Physik, München 1
Outline I. Foundations 1. String theory 2. Dualities between quantum field theories and gravity theories AdS/CFT correspondence 3. Originates from string theory II. Applications 1. Theory of Strong Interactions (QCD) 2. Quark-gluon plasma and black holes 3. Superfluidity and superconductivity 2
Well-known fundamental theories: Introduction: String theory
Introduction: String theory Well-known fundamental theories: Standard model of elementary particle physics: describes strong, weak and electromagnetic forces Local quantum field theory (gauge theory), particles point-like Confirmed experimentally up to a scale of about O(100) GeV Effective Theory (free parameters, for example quark masses)
Introduction: String theory Well-known fundamental theories: Standard model of elementary particle physics: describes strong, weak and electromagnetic forces Local quantum field theory (gauge theory), particles point-like Confirmed experimentally up to a scale of about O(100) GeV Effective Theory (free parameters, for example quark masses) Einstein s theory of general relativity (Theory of gravitation) Classical theory Non-renormalizable Quantum effects expected at M P lanck 10 19 GeV 3
Introduction: String theory Search for a Unified Theory of fundamental interactions Quantum theory of gravitation Description of all four interactions in a unified framework Reduction of the number of free parameters Promising candidate: String theory 4
Introduction: String theory Quantum theory of gravitation and unification of all four interactions: Locality is abandoned at very short distances Natural cut-off: String length l s 1 M P lanck, l s = G c 3 = 1.616 10 35 m
Introduction: String theory Quantum theory of gravitation and unification of all four interactions: Locality is abandoned at very short distances Natural cut-off: String length l s 1 M P lanck, l s = G c 3 = 1.616 10 35 m Direct experimental tests not possible today
Introduction: String theory Quantum theory of gravitation and unification of all four interactions: Locality is abandoned at very short distances Natural cut-off: String length l s 1 M P lanck, l s = G c 3 = 1.616 10 35 m Direct experimental tests not possible today Search for indirect experimental evidence 5
New applications of string theory New: String theory: New framework for describing strongly coupled systems
New applications of string theory New: String theory: New framework for describing strongly coupled systems At arbitrary energies
New applications of string theory New: String theory: New framework for describing strongly coupled systems At arbitrary energies New applications of string theoretical methods within elementary particle physics within condensed matter theory 6
Dualities between quantum field theories and gravitation Map: Quantum field theory Classical theory of Gravitation Event at the RHIC accelerator Black hole geometry
Dualities between quantum field theories and gravitation Map: Quantum field theory Classical theory of Gravitation Event at the RHIC accelerator Black hole geometry Practical Significance: Map: Strongly coupled quantum field theory Weakly coupled classical theory of gravitation (Solvable) 7
AdS/CFT Correspondence: Foundations (Maldacena 1997, AdS: Anti de Sitter space, CFT: conformal field theory) Follows from a low-energy limit of string theory Duality: Quantum field theory at strong coupling Theory of gravitation at weak coupling Holography: Quantum field theory in four dimensions Gravitational theory in five dimensions 8
Anti-de Sitter Space Space of constant negative curvature, has a boundary ds 2 = e 2r/L dx µ dx µ + dr 2 Figure source: Institute of Physics, Copyright: C. Escher 9
Quantum field theory Conformal field theory
Conformal field theory Quantum field theory in which the fields transform covariantly under conformal transformations
Conformal field theory Quantum field theory in which the fields transform covariantly under conformal transformations Conformal coordinate transformations: preserve angles locally
Conformal field theory Quantum field theory in which the fields transform covariantly under conformal transformations Conformal coordinate transformations: preserve angles locally Correlation functions are determined up to a small number of parameters
Conformal field theory Quantum field theory in which the fields transform covariantly under conformal transformations Conformal coordinate transformations: preserve angles locally Correlation functions are determined up to a small number of parameters Note: Confinement and conformal symmetry are incompatible! 10
String theory Open strings : Gauge degrees of freedom of the Standard Model Closed Strings: Gravitation Two types of degrees of freedom: open and closed strings Higher oscillation modes are excited 11
String theory Quantization: Supersymmetric string theory well-defined in 9 + 1 dimensions (No tachyons, no anomalies) Supersymmetry: Symmetry Bosons Fermions What is the significance of the additional dimensions? 1. Compactification 2. D-branes 12
D-Branes D-branes are hypersurfaces embedded into 9+1 dimensional space D3-Branes: (3+1)-dimensional hypersurfaces Open Strings may end on these hypersurfaces Dynamics 13
D-Branes Low-energy limit (Strings point-like) Open Strings Dynamics of gauge fields on the brane
D-Branes Low-energy limit (Strings point-like) Open Strings Dynamics of gauge fields on the brane Second interpretation of the D-branes: Solitonic solutions of ten-dimensional supergravity Heavy objects which curve the space around them 14
String theory origin of the AdS/CFT correspondence D3 branes in 10d duality AdS x S5 5 near-horizon geometry Low energy limit Supersymmetric SU(N) gauge theory in four dimensions (N ) Supergravity on the space AdS 5 S 5 15
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Applications 1. Strong interactions - Quarks and Gluons 2. Finite temperature: Quark Gluon Plasma 3. Superfluiditiy and Superconductivity 17
Strong Interactions: Quarks and Gluons Theory of Strong Interactions: Quantum Chromodynamics (QCD) Baryon (e.g. Proton) Meson (e.g. Pion) q q q q q Asymptotic freedom and confinement: Quarks and Gluons are strongly coupled at low energies 18
Quarks in the AdS/CFT correspondence Add D7-Branes (Hypersurfaces) 0 1 2 3 4 5 6 7 8 9 N D3 X X X X 1,2 D7 X X X X X X X X Quarks: Low energy limit of open strings between D3- and D7-Branes Meson masses from fluctuations of the hypersurface (D7-Brane):
Quarks in the AdS/CFT correspondence Add D7-Branes (Hypersurfaces) 0 1 2 3 4 5 6 7 8 9 N D3 X X X X 1,2 D7 X X X X X X X X Quarks: Low energy limit of open strings between D3- and D7-Branes Meson masses from fluctuations of the hypersurface (D7-Brane): Confinement: AdS 5 S 5 -metric has to be deformed 19
Comparison to lattice gauge theory Mass of ρ meson as function of π meson mass 2 (for N ) m Ρ 1 0.9 0.8 0.8 0.7 0.6 m ρ 0.6 0.4 0.2 0.5 0.4 SU(2) SU(3) SU(4) SU(6) N = inf. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m Π 2 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 2 m π J.E., Evans, Kirsch, Threlfall 07, review EPJA Lattice: Lucini, Del Debbio, Patella, Pica 07 AdS/CFT result: m ρ (m π ) m ρ (0) = 1 + 0.307 ( ) 2 mπ m ρ (0) Lattice result (from Bali, Bursa 08): slope 0.341 ± 0.023 20
2. Finite temperature and density Prime example: Phase diagram of strongly interacting matter Bild: CBM @ FAIR, GSI 21
2. : Finite Temperature and Chemical Potential Quark-gluon plasma: Strongly coupled state of matter above deconfinement temperature T d
2. : Finite Temperature and Chemical Potential Quark-gluon plasma: Strongly coupled state of matter above deconfinement temperature T d AdS/CFT dual of field theory at finite temperature: AdS black hole Hawking temperature temperature in the dual quantum field theory 22
Universal result: Quark-gluon plasma
Quark-gluon plasma Universal result: Ratio of shear viscosity and entropy η s = k B 1 4π
Quark-gluon plasma Universal result: Ratio of shear viscosity and entropy η s = 1 k B 4π Calculated using AdS/CFT at finite temperature Policastro, Son, Starinets 2001
Quark-gluon plasma Universal result: Ratio of shear viscosity and entropy η s = 1 k B 4π Calculated using AdS/CFT at finite temperature Policastro, Son, Starinets 2001 Lower bound: smallest possible value 23
2. Quarks and mesons at finite temperature Babington, J.E., Evans, Guralnik, Kirsch 2004; Mateos, Myers, Thomson 2006
2. Quarks and mesons at finite temperature Babington, J.E., Evans, Guralnik, Kirsch 2004; Mateos, Myers, Thomson 2006 Prediction: Mesons survive deconfinement if quark mass m q sufficiently large 1st order fundamental phase transition at T = T f where mesons dissociate T f 0.12M meson at T =0 24
Finite baryon chemical potential Easy to introduce: VEV for time component of gauge field ρ vector meson spectral function in dense hadronic medium 4 Im G/(Nf Nc T 2 ) 3000 2500 2000 1500 m = 5 d = 0.5 d = 1 d = 2 d = 3 1000 500 0 3.5 4.0 4.5 5.0 5.5 6.0 ω/(2πt ) AdS/CFT result (J.E., Kaminski, Kerner, Rust 2008) 25
Spectral function at finite baryon density ρ vector meson spectral function in dense hadronic medium 4 Im G/(Nf Nc T 2 ) 3000 2500 2000 1500 1000 m = 5 d = 0.5 d = 1 d = 2 d = 3 ImD ρ [GeV 2 ] 10 8 6 4 vacuum ρ N =0.5ρ 0 ρ N =1.0ρ 0 ρ N =2.0ρ 0 q=0.3gev 500 2 0 3.5 4.0 4.5 5.0 5.5 6.0 AdS/CFT result ω/(2πt ) (J.E., Kaminski, Kerner, Rust 2008) 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 M [GeV] Field theory (Rapp, Wambach 2000) 26
3. Superfluidity and Superconductivity Chemical Potential and Finite Density for Isospin (SU(2)) u- and d-quarks Phase diagram Instability! 27
3. Superfluidity and Superconductivity
3. Superfluidity and Superconductivity New solution to the equations of motion with lower free energy
3. Superfluidity and Superconductivity New solution to the equations of motion with lower free energy The new solution contains a condensate ψ u γ 3 ψ d + ψ d γ 3 ψ u + bosons
3. Superfluidity and Superconductivity New solution to the equations of motion with lower free energy The new solution contains a condensate ψ u γ 3 ψ d + ψ d γ 3 ψ u + bosons ρ meson condensate (p-wave, triplet pairing) Ammon, J.E., Kaminski, Kerner 2008 28
Superfluidity and Superconductivity The new ground state is a superfluid. Frequency-dependent conductivity from spectral function Ê Û 1.4 1.2 1.0 0.8 ÒØ 0.6 0.4 0.2 Ì ¼ ¼Ì Ì ¼ Ì Ì ¼ Ì Ì ¼ ¾ Ì 0.0 0 5 10 15 20 Û w = ω/(2πt ) Prediction: Frictionless motion of mesons through the plasma 29
Quantum Phase Transition Two chemical potentials: Isospin and Baryon Chemical Potential 0.35 0.30 Â ½ Ü ¼ Õ ¾ Ì Á 0.25 0.20 0.15 Â ½ Ü ¼ 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Á Example for Quantum Phase Transition Figure by Patrick Kerner 30
Quantum Phase Transition Phase diagrams Quantum phase transition Superconductor 31
Fermionic Excitations in p-wave Superconductors Ammon, J.E., Kaminski, O Bannon 2010 32
B-field induced rho meson condensation Field theory calculation (Chernodub) Gauge/gravity calculation (Bu, J.E., Shock, Strydom) 33
Outlook: Universality and Homes Law
Outlook: Universality and Homes Law Universal result from gauge/gravity duality for hydrodynamics: η s = k B 1 4π
Outlook: Universality and Homes Law Universal result from gauge/gravity duality for hydrodynamics: η s = 1 k B 4π Search for a similar relation in condensed matter physics
Outlook: Universality and Homes Law Universal result from gauge/gravity duality for hydrodynamics: η s = 1 k B 4π Search for a similar relation in condensed matter physics Candidate: Homes Law for superconductors: n s (T = 0) = Cσ(T c )T c n s : Superfluid density; σ: Conductivity
Outlook: Universality and Homes Law Universal result from gauge/gravity duality for hydrodynamics: η s = 1 k B 4π Search for a similar relation in condensed matter physics Candidate: Homes Law for superconductors: n s (T = 0) = Cσ(T c )T c n s : Superfluid density; σ: Conductivity Found experimentally to great accuracy 34
Outlook: Homes Law and Universality Relation between η/s and Homes Law:
Outlook: Homes Law and Universality Relation between η/s and Homes Law: Shortest possible dissipation time in strongly coupled systems: Planckian time: τ = k B T c
Outlook: Homes Law and Universality Relation between η/s and Homes Law: Shortest possible dissipation time in strongly coupled systems: Planckian time: τ = k B T c Work in progress: Obtain this relation within gauge/gravity duality 35
Conclusion and Outlook Generalized AdS/CFT correspondence: Gauge/Gravity Duality New relation between fundamental and applied aspects of theoretical physics
Conclusion and Outlook Generalized AdS/CFT correspondence: Gauge/Gravity Duality New relation between fundamental and applied aspects of theoretical physics Identification of Universal Behaviour
Conclusion and Outlook Generalized AdS/CFT correspondence: Gauge/Gravity Duality New relation between fundamental and applied aspects of theoretical physics Identification of Universal Behaviour Predictions for mesons in the quark-gluon plasma
Conclusion and Outlook Generalized AdS/CFT correspondence: Gauge/Gravity Duality New relation between fundamental and applied aspects of theoretical physics Identification of Universal Behaviour Predictions for mesons in the quark-gluon plasma Quantum phase transitions
Conclusion and Outlook Generalized AdS/CFT correspondence: Gauge/Gravity Duality New relation between fundamental and applied aspects of theoretical physics Identification of Universal Behaviour Predictions for mesons in the quark-gluon plasma Quantum phase transitions Comparison and combination with other approaches New relations between different areas of physics 36