Warped AdS 3 black holes in higher derivative gravity theories C - PowerPoint PPT Presentation

Introduction Method Warped duality Entropies in the warped duality Conclusion Warped AdS 3 black holes in higher derivative gravity theories C´ eline Zwikel November 17, 2016 C´ eline Zwikel 1 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Introduction Context: Holographic dualities in 2+1 dimensions in this talk: warped duality Main question: Bulk/Boundary entropy matching in arbitrary higher derivative theory of gravity C´ eline Zwikel 2 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion AdS 3 /CFT 2 In general relativity (GR), Strominger ’97 shows S BTZ = S CFT Cardy . BH Indeed, = π r + = π �� � � S BTZ 2 ℓ ( ℓ M + J ) + 2 ℓ ( ℓ M − J ) BH 2 2 � � c ¯ c L 0 L 0 S CFT Cardy = 2 π + 2 π 6 6 ℓ ( L 0 + ¯ L 0 ) and J = L 0 − ¯ One has M = 1 L 0 . C´ eline Zwikel 3 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Higher Derivative (HD) or Higher Curvature (HC) General Lagrangian (diffeomorphism invariant without gravitational anomalies): L = ⋆ f ( g ab , R abcd , ∇ e 1 R abcd , ∇ ( e 1 ∇ e 2 ) R abcd , ..., ∇ ( e 1 ... ∇ e n ) R abcd ) Example: New Massive Gravity 1 d 3 x √− g ( R − 2 Λ) + 1 R µν R µν − 3 � � �� � 8 R 2 L NMG = m 2 16 π Why? GR = low energy effective action of an UV compete theory (ex:ST) Corrections to the Einstein-Hilbert action are expected. C´ eline Zwikel 4 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Match of the entropies in HD? Modification of the entropies Bulk: Bekenstein-Hawking → Iyer-Wald The entropy law is modified to keep the first law valid. Boundary Entropy formula depends of the charges who are theory dependant. A priori no reason that the match is still preserved. C´ eline Zwikel 5 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion AdS 3 /CFT 2 in a HD theory Effect of higher curvature terms boils down in a global multiplicative renormalization of all charges of the theory. Ref: Saida-Soda ’99, Kraus-Larsen ’05 Bulk: S HD = α S GR Boundary  � �  c HC L HC c HC ¯ L HC S CFT , HC = 2 π 0 0 +   6 6  �  � c ¯ c L 0 L 0  = α S CFT , GR = 2 πα + 2 π  6 6 C´ eline Zwikel 6 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion AdS 3 /CFT 2 in a HD theory Effect of higher curvature terms boils down in a global multiplicative renormalization of all charges of the theory. Ref: Saida-Soda ’99, Kraus-Larsen ’05 Bulk: S HD = α S GR Boundary S CFT , HD = α S CFT , GR C´ eline Zwikel 6 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion AdS 3 /CFT 2 in a HD theory Effect of higher curvature terms boils down in a global multiplicative renormalization of all charges of the theory. Ref: Saida-Soda ’99, Kraus-Larsen ’05 Bulk: S HD = α S GR Boundary S CFT , HD = α S CFT , GR Consequently, S HD = S CFT , HD . C´ eline Zwikel 6 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Plan 1. Method 2. Warped duality 3. Entropies in the warped duality C´ eline Zwikel 7 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Method C´ eline Zwikel 8 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Method The methods used in AdS 3 /CFT 2 are not transposable to our case of interest. Inspired by the work of Azeyanagi, Comp` ere, Ogawa, Tachikawa and Terashima (arXiv: 0903.4176) on Bulk/Boundary entropy match for 4D Extremal BH Formalism: Covariant phase formalism C´ eline Zwikel 9 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Covariant Phase Space Formalism L : Lagrangian n -form The EOM E = 0 are determined through δ L (Φ) = E (Φ) δ Φ + d Θ [ δ Φ , Φ] with Θ [ δ Φ , Φ] : symplectic potential ( n − 1-form). The symplectic structure of the configuration phase � Ω [ δ 1 Φ , δ 2 Φ; Φ] = ω [ δ 1 Φ , δ 2 Φ; Φ] C in terms of the symplectic current ω = δ Θ . C´ eline Zwikel 10 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Invariance of the Lagrangian under diffeomorphisms δ ξ L = L ξ L = d ( i ξ L ) . Thus, E (Φ) δ Φ = d ( Θ [ δ Φ , Φ] − i ξ L ) . The Noether current is defined as J (Φ) := Θ [ δ Φ , Φ] − i ξ L . J is closed on-shell: d J ≈ 0. Thus, it exists a n -form Q, called the Noether charge s.t. J ξ (Φ) := − d Q ξ (Φ) . C´ eline Zwikel 11 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion One defines k ξ ( δ Φ , Φ) := − i ξ Θ [ δ Φ , Φ] − δ Q ξ (Φ) . If Φ satisfy EOM, δ Φ the linearized EOM, ω [ δ ξ Φ , δ Φ; Φ] = dk ξ ( δ Φ , Φ) . Integrating that equation, � δ H ξ = Ω [ δ ξ Φ , δ Φ; Φ] = k ξ ( δ Φ , Φ) Σ= ∂ C where H ξ is the Hamiltonian generating the flow Φ → δ ξ Φ . C´ eline Zwikel 12 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion One can show that the representation of asymptotic symmetry algebra by a Dirac bracket is � δ ξ H ζ := { H ζ , H ξ } = H [ ζ,ξ ] + k ζ ( δ ξ Φ , Φ) . Σ= ∂ C � So the central charge is Σ= ∂ C k ζ ( δ ξ Φ , Φ) . C´ eline Zwikel 13 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Ambiguities in the definitions : In the literature, it was advocated that the so-called invariant symplectic structure, based on cohomological argument is the one to be used. ω inv = ω − dE k inv ( δ Φ , Φ) = k ξ ( δ Φ , Φ) − E ( δ Φ , Φ) ξ In our cases, this E term will always be zero. We can only considered the symplectic formulation. C´ eline Zwikel 14 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Higher curvature Lagragian (without derivative) Higher curvature Lagragian L = ⋆ f ( g ab , R abcd ) Rewritten in terms of auxiliary fields Z and R abcd L = ⋆ [ f ( g ab , R abcd ) + Z abcd ( R abcd − R abcd )] The EOM for the auxiliary fields are Z abcd = ∂ f ( g ab , R abcd ) , R abcd = R abcd . ∂ R abcd C´ eline Zwikel 15 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion L = ⋆ [ f ( g ab , R abcd ) + Z abcd ( R abcd − R abcd )] No higher than the second derivative of g ab in the Lagrangian, so we can derive, for example in 3D � − Z abcd ∇ c ξ d − 2 ξ c ∇ d Z abcd � ( Q ξ ) c = ǫ abc � Z abcd ∇ d δ g bc − δ g bc ∇ d Z abcd � Θ ef = − 2 ǫ aef . C´ eline Zwikel 16 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Exact charges For an axisymmetric and stationary BH, the mass and the angular momentum are defined � � δ M HC := δ H ∂ t = δ Q ∂ t + i ∂ t Θ ∞ ∞ � δ J HC := δ H ∂ φ = − δ Q ∂ φ ∞ And the Iyer-Wald entropy, � S HC = S IW = − 2 π dA Z abcd ǫ ab ǫ cd horizon where ǫ ab is the binormal to the horizon. C´ eline Zwikel 17 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Warped duality C´ eline Zwikel 18 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Bulk Warped AdS 3 � � 2 � ℓ 2 4 ν 2 ds 2 = − cosh 2 ( σ ) d τ 2 + d σ 2 + � du + sinh ( σ ) d τ ν 2 + 3 ν 2 + 3 Isometry group: SL ( 2 , R ) × U ( 1 ) ν = 1: AdS C´ eline Zwikel 19 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Warped BHs Spacelike stretched warped black holes are dr 2 − 4 ν r � � ds 2 = dt 2 + + dt d φ ( ν + 3 ) r 2 − 12 mr + 4 j ℓ ℓ ℓ ν 3 ( ν 2 − 1 ) � � r 2 + 12 mr − 4 j ℓ d φ 2 ℓ 2 ν m , j : parameters characterising the BH ( j < 9 ℓ m 2 ν with 3 + ν 2 ) ℓ : original AdS radius ν 2 > 1 . Locally warped AdS 3 Solutions of New Massive Gravity (not GR) C´ eline Zwikel 20 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Boundary Warped CFT (WCFT) Chiral scaling symmetry on 2D FT → extended local algebra Virasoro-Kac-Moody U ( 1 ) i [ L m , L n ] = ( m − n ) L m + n + c 12 ( m 3 − m ) δ m + n , 0 i [ L m , P n ] = − nP m + n i [ P m , P n ] = k 2 m δ m + n , 0 . Partition function Z ( β, θ ) = Tr e − β P 0 − β Ω L 0 Modular covariance → Cardy-like entropy formula − 8 π 2 S WCFT = 2 π i Ω P vac β Ω L vac 0 0 C´ eline Zwikel 21 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Entropies in the warped duality C´ eline Zwikel 22 / 30

Introduction Method Warped duality Entropies in the warped duality Conclusion Form of a generic tensor made out of the metric Property of maximally symmetric spacetimes (ex: AdS 3 ), all the tensors made out of the curvature and its covariant derivatives can be expressed in terms of the metric. Ex: R µν = − 2 ℓ 2 g µν C´ eline Zwikel 23 / 30