AdS/CFT and Lovelock Gravity Manuela Kulaxizi University of Uppsala - PowerPoint PPT Presentation

AdS/CFT and Lovelock Gravity Manuela Kulaxizi University of Uppsala

Introduction • The AdS/CFT correspondence provides a tool for studying large N c gauge theories at strong coupling. Has been applied to several problems of interest from nuclear physics to condensed matter (chiral symme- try breaking, viscosity to entropy ratio, marginal fermi liquid description, superconductors etc.) • Interesting to study higher derivative gravity theo- ries in the context of the AdS/CFT correspondence. They provide a holographic example where c � = a .

Introduction Gravitational theories with higher derivative terms in general • Have ghosts when expanded around flat space. • Their equations of motion contain more than two derivatives of the metric. Hard to solve exactly. Additional degrees of freedom. In holography, this implies the existence of extra op- erators in the boundary CFT. [Skenderis, Taylor and van Rees].

Introduction There exists a special class of gravitational theories with higher derivative terms, Lovelock gravity. [ d � 2 ] d d +1 x √− g � ( − ) p ( p − 2 d )! S = ( p − 2)! λ p L p p =0 with [ d d 2 ] the integral part of 2 , λ p are the Lovelock pa- rameters and the p -th order Lovelock term L p is L p = 1 2 p δ µ 1 ν 1 ··· µ p ν p ρ 1 σ 1 ··· ρ p σ p R ρ 1 σ 1 µ 1 ν 1 · · · R ρ p σ p µ p ν p L p is the Euler density term in 2 p –dimensions.

Introduction We choose λ 0 = 1 and λ 1 = − 1 such that L 0 = d ( d − 1) L 1 = R . L 2 Examples: • 2nd order Lovelock term ⇔ Gauss-Bonnet L 2 = R 2 µνρσ − 4 R 2 µν + R 2 • 3rd order Lovelock term σν R µν L 3 = 2 R ρσκλ R κλµν R µν ρσ + 8 R ρσ κµ R κλ ρλ + ρ + 3 RR 2 + 24 R ρσκλ R κλσµ R µ ρσκλ + 24 R ρκσλ R σρ R λκ + ρ − 12 RR 2 ρσ + R 3 + 16 R ρσ R σκ R κ

Introduction Special Properties of the Lovelock action: • Equations of motion contain only up to second order derivatives of the metric ⇒ No additional boundary data. Black hole solutions can be found exactly . • No ghosts when expanded around Minkowski flat back- ground. • Palatini and Metric formulations equivalent [Exirifard, Sheikh–Jabbari].

Introduction Lovelock gravity admits AdS solutions with radius L 2 AdS = αL 2 where α = α ( λ p ) Example: Gauss-Bonnet term λ 2 � = 0 � √ � α = 1 1 + 1 − 4 λ 2 2 Asymptotically AdS black hole solutions exist d − 1 ds 2 = − f ( r ) dt 2 + dr 2 � f ( r ) + r 2 dx 2 i i =1 where f ( r ) satisfies the equation of motion   ′ � f � p � r + � d � � ( d − 1) λ p r d − 2 p f p  = 0 ⇒ λ p = r 2 r p p

Introduction Study Lovelock theories of gravity in the context of the AdS/CFT correspondence. What new features does the boundary CFT acquire given the additional param- eters of the theory λ p ? Can we learn something new? In this talk: Part 1: Energy Flux Positivity ⇒ Absence of Ghosts Part 2: Focus on holographic entanglement entropy. New features and tests [work in progress].

Outline • Part 1. – Review of causality and energy flux positivity cor- respondence – Absence of ghosts and energy flux positivity in field theory. • Part 2. – Entanglement Entropy: A review – EE in four dimensional CFTs : Solodukhin’s Result – Holographic Description of Entanglement Entropy – Fursaev’s proposal and Generalizations – Summary, Conclusions and Open Questions

Part 1. Absence of ghosts and Positivity of the Energy Flux

Fluctuation Analysis Study quasinormal modes of the AdS black hole solution ⇒ Pole Structure of the retarded stress-energy tensor two point function. • Consider metric fluctuations δg 12 = φ ( r, t, x d − 1 ) Corresponds to � T 12 ( x ) T 12 (0) � (scalar channel). • Perform a Fourier Transform � dωdq (2 π ) 2 ϕ ( r ) e − iωt + iqx d − 1 , k = ( ω, 0 , 0 , · · · , 0 , q ) . φ ( t, r, x d − 1 ) = Express the equation of motion for ϕ in Schrodinger form � � Ψ = ω 2 − 1 g ( y ) + V 1 ( y ) q 2 ∂ 2 c 2 y Ψ + q 2 Ψ q 2 The horizon is now at y = −∞ and the boundary at y = 0 whereas Ψ ∼ ϕ .

Fluctuation Analysis What is the behavior of the potential? V 1 ( y ) is monotonically increasing function. Monotonicity properties of c 2 g ( y ) depend on λ p . It is either monotonically increasing, reaching maximum at the boundary c 2 g = 1 , or develops a maximum in the bulk c 2 g,max > 1 and metastable states may appear in the spectrum.

Fluctuation Analysis Consider the large q limit. Replace V 1 ( y ) by an infinite wall at y = 0 . Use the WKB approximation to determine the group velocity of the states in the dual CFT. U = dω dq → c 2 g,max Conclusion: For values of the Lovelock parameters λ p such that c 2 g ( y ) at- tains a maximum greater than unity in the bulk, the boundary theory contains superluminal states, i.e., violates causality. Method by [Brigante, Liu, Myers, Shenker, Yaida].

Causality Bounds The specific form of the constraints on the Lovelock parameters λ p are determined by the near boundary be- havior of c 2 g g = 1 − C ( λ p ) r d + c 2 r d + · · · where � p p (( d − 2)( d − 3) + 2 d ( p − 1)) λ p α p − 1 C ( λ p ) = − �� � 2 p pλ p α p − 1 α ( d − 2) ( d − 3) Preserving causality in the dual theory � p (( d − 2)( d − 3) + 2 d ( p − 1)) λ p α p − 1 < 0 C ( λ p ) ≥ 0 ⇒ p [de Boer, Parnachev, M.K.] [Buchel, Escobedo, Myers, Paulos, Sinha, Smolkin] [Camanho, Edelstein]

Causality Bounds Similar results can be obtained from studying graviton perturbations of different helicity. Each polarization gives a different constraint: C 1 ( λ p ) > 0 , C 2 ( λ p ) > 0 , C 3 ( λ p ) > 0 [Myers, Buchel; Hofman; Camanho, Edelstein]. Examples: Gauss–Bonnet gravity d = 4 : − 7 9 35 < λ 2 < 100 3rd order Lovelock gravity d = 6 : C ( λ p ) = α 5 α 2 λ 2 + (9 − 8 α ) [ α 2 λ 2 + (3 − 2 α )] 2 ≥ 0

Positivity of the Energy Flux • What do the Lovelock parameters λ p correspond to in the boundary CFT? What are the corresponding constraints? The two- and three-point functions of the stress energy tensor are completely determined up to three indepen- dent coefficients ( A , B , C ) [Osborn, Petkou]. � T µν ( x ) T ρσ (0) � = ( d − 1)( d + 2) A − 2 B − 4( d + 1) C I µν,ρσ ( x ) x 2 d d ( d + 2) � T µν ( x 3 ) T ρσ ( x 2 ) T τκ ( x 1 ) � = AJ µνρστκ ( x ) + BK µνρστκ ( x ) + x d 12 x d 13 x d x d 12 x d 13 x d 23 23 + CM µνρστκ ( x ) x d 12 x d 13 x d 23

Positivity of the Energy Flux The Lovelock parameters λ p can be expressed in terms of the CFT parameters A , B , C . Then holography pre- dicts that A , B , C obey three independent constraints: C 1 ( A , B , C ) > 0 , C 2 ( A , B , C ) > 0 , C 3 ( A , B , C ) > 0 These constraints precisely match the constraints de- rived from the positivity of the energy flux one-point function! [Hofman, Maldacena] Note: Supersymmetry implies a linear relation between A , B , C . Effectively, two independent parameters. Example: the central charges a, c in d = 4 . Curiously, the Lovelock parameters satisfy this relation.

Positivity of the Energy Flux Definition: The energy flux operator E ( � n ) per unit angle measured through a very large sphere of radius r is � n i T 0 r →∞ r d − 2 n i ) E ( � n ) = lim i ( t, r � dt � n i is a unit vector specifying the position on S d − 2 where energy measurements may take place. Integrating over all angles yields the total energy flux at large distances. Focus on the energy flux one-point function on states created by the stress–energy tensor operator O q = ǫ ij T ij ( q ) with ǫ ij a symmetric, traceless polarization tensor

Positivity of the Energy Flux • Rotational symmetry fixes the form of the energy flux one–point function up to two independent parame- ters. n ) � T ij = � ǫ ∗ ik T ik E ( � n ) ǫ lj T lj � �E ( � = � ǫ ∗ ik T ik ǫ lj T lj � � � � � �� ǫ ∗ | ǫ ij n i n j | 2 1 2 E il ǫ lj n i n j = 1 + t 2 − + t 4 − d 2 − 1 ǫ ∗ ǫ ∗ Ω d − 2 d − 1 ij ǫ ij ij ǫ ij By construction t 2 , t 4 can be expressed in terms of the CFT parameters A , B , C . The supersymmetric case: the linear relation between A , B , C is equivalent t 4 = 0 .

Positivity of the Energy Flux Demand positivity of the energy flux one point function, i.e. , �E ( � n ) � ≥ 0 . The positivity of the energy flux imposes constraints on t 2 , t 4 : 1 2 C 1 ( A , B , C ) ≡ 1 − d − 1 t 2 − d 2 − 1 t 4 ≥ 0 1 2 d 2 − 1 t 4 + t 2 C 2 ( A , B , C ) ≡ 1 − d − 1 t 2 − 2 ≥ 0 1 d 2 − 1 t 4 + d − 2 2 C 3 ( A , B , C ) ≡ 1 − d − 1 t 2 − d − 1( t 2 + t 4 ) ≥ 0 When expressed in terms of A , B , C these constraints pre- cisely match the ones obtained from holography!

Example: Bounds for a d = 6 dimensional SCFT Parameter space t 2 , t 4 of a consistent CFT. Values out- side the triangle are forbidden.

Absence of ghosts and CFT constraints The energy flux positivity constraints are related to causality in the gravity language. Can we see some- thing similar in field theory? Guide from the AdS/CFT analysis: • Consider the Fourier transform of the two–point func- tion of the stress energy tensor at finite temperature. • Three independent polarizations; each polarization yields a different set of constraints. k • Focus on large momenta, small temperatures T ≫ 1 .