Outline Higher derivative gravity and AdS/CFT Interacting fermions and AdS/CFT Some extentions of the AdS/CFT correspondence Andrei Parnachev Leiden University May 5, 2011 Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Higher derivative gravity and AdS/CFT Interacting fermions and AdS/CFT Introduction There are many ways to modify the original AdS/CFT correspondence. Here I will talk about two examples. Example 1: Keeping the theory conformal but modifying 3-pt functions of T ab . Achieved by considering higher derivative gravity in the bulk. Example2: Introducing small number of defect fermions interacting with CFT. Leads to tachyon condensation in the bulk. Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Higher derivative gravity and AdS/CFT Interacting fermions and AdS/CFT Outline Higher derivative gravity and AdS/CFT Introduction to higher derivative gravities Lovelock gravities Implications for field theories Interacting fermions and AdS/CFT Introduction D3/D7 at strong coupling Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Introduction to higher derivative gravities Gauss-Bonnet gravity Lagrangian: L = R + 6 L 2 + λ L 2 ( R 2 − 4 R ab R ab + R abcd R abcd ) We will be interested in the λ ∼ 1 regime. More generally, we can add terms O ( R k ) which are Euler densities in 2 k dimensions: λ k L 2 k − 2 δ a 1 ... b k c 1 ... d k R c 1 d 1 a 1 b 1 . . . R c k d k a k b k They become non-trivial for gravity theories in AdS D with D > 2 k . Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Special properties ◮ Equations of motion don’t contain 3rd order derivatives g ′′′ ◮ Metric and Palatini formulations are equivalent ◮ No ghosts around flat space ◮ Exact black hole solutions can be found The last property allows one to study dual CFTs at finite temperature. The first property implies that holographic dictionary is not modified. Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Implications for AdS/CFT The UV behavior is modified. Consider conformal rescaling g CFT → exp(2 σ ) g CFT ab ab CFT action is anomalous; there are two terms in 3+1 dimensions: δ W = aE 4 + cW 2 Enstein-Hilbert (E-H) implies a = c . Lovelock implies a � = c . Of course, the IR behavior is modified as well. E.g. values of transport coefficients are different from E-H. Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Finite T; metastable states Consider propagation of gravitons in the black hole background. (D=5 Gauss-Bonnet: Brigante, Liu, Myers, Shenker, Yaida) α dt 2 + dr 2 f ( r )+ r 2 ds 2 = − f ( r ) �� � dx 2 i +2 φ ( t , r , z ) dx 1 dx 2 L 2 � Fourier transform: φ ( t , r , z ) = dwdqexp ( − iwt + iqz ) After substitutions and coordinate transformations, get Schrodinger equation with � → 1 / ˜ q = T / q : y Ψ( y ) + V ( y )Ψ( y ) = w 2 − 1 q 2 ∂ 2 q 2 Ψ( y ) ˜ Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Causality Spectrum = states in finite T CFT. In the ˜ q ≫ 1 regime, there are stable states with ∂ w /∂ q > 1 in some region of the parameter space. Causality places constraints on Lovelock couplings: [( d − 2)( d − 3) + 2 d ( k − 1)] λ k α k − 1 < 0 � k where α defines the AdS radius L 2 AdS = L 2 /α and satisfies k λ k α k = 0. � This effect is absent at T = 0; appears as O ( T / q ) correction from the tails of black hole metric. Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Positivity of energy flux = unitarity n ) = lim r →∞ r 2 � dtT 0 Define ε (ˆ i ˆ n i Conjecture (Hofman, Maldacena) � ε (ˆ n ) � ≥ 0. Consider a state created by ǫ ij T ij n j ) 2 n ) �∼ 1+ t 2 ( ǫ ij ǫ il ˆ n j ˆ d − 1)+ t 4 (( ǫ ij ˆ n i ˆ n l 1 2 � ε (ˆ − − d 2 − 1) ǫ ij ǫ ij ǫ ij ǫ ij t 2 and t 4 are determined by the 2 and 3-point functions of T ab . ◮ energy flux positivity in CFTs dual to Lovelock gravities is equivalent to causality at finite temperature! ◮ also equivalent to the absence of ghosts (at finite temperature). Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Holographic entanglement entropy Consider two systems A , B with Hilbert spaces consisting of two states {| 1 � , | 2 �} . Reduced density matrix of A is obtained by tracing over B; entanglement entropy is the resulting VN entropy. ρ A = tr B ρ ; S A = − tr A ρ A log ρ A Product state: | 1 A 1 B � ⇒ S A = 0 Pure (non product) state: 1 √ ( | 1 A 2 B � − | 2 A 1 B � ) ⇒ S A = ln 2 2 Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Holographic EE Consider EE in CFT dual to Lovelock gravity in AdS . A proposal for holographic EE (Fursaev) . √ σ 1 � 1 + λ 2 L 2 R Σ � � S ( V ) = G (5) Σ N Σ is the minimal surface ending on ( ∂ V ) which satisfies the e.o.m. derived from this action. R Σ is the induced scalar curvature on Σ. Consider the case of a ball, bounded by the two-sphere of radius R . It is not hard to solve EOM near the boundary of AdS and extract the log-divergent term: S ( B ) = R 2 ǫ 2 + a 90 ln R /ǫ + . . . Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories a-theorem and AdS/CFT Zamolodchikov’s c-theorem in 1+1 dimensions: there is a c-function [made out of � T ab T ab � ] which is ◮ positive ◮ decreases along the RG flows ◮ equal to the central charge c at fixed points Is there an analogous quantity in 3+1 dimensions? Conjecture: yes, and it is equal to a at fixed points. Holographic a-theorem (Myers, Sinha) Consider a background which holographically describes the RG flow: ds 2 = exp(2 A ( r ))( dx µ ) 2 + dr 2 Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories a-theorem and AdS/CFT Then the quantity 1 1 − 6 λ L 2 A ′ ( r ) 2 � � a ( r ) = l 3 p A ′ ( r ) 3 is equal to a at fixed points and satisfies 1 T 0 0 − T r a ′ ( r ) = − � � r l 3 p A ′ ( r ) 4 The right hand side of this equation is proportional to the null energy condition. Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Comments We have seen that adding higher curvature terms to the usual AdS/CFT setup gives useful insight for the physics of CFTs. Other directions include ◮ Applications in non-relativistic holography Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Comments We have seen that adding higher curvature terms to the usual AdS/CFT setup gives useful insight for the physics of CFTs. Other directions include ◮ Applications in non-relativistic holography ◮ Tests for supersymmetrizability of higher derivative gravities (HDG) Andrei Parnachev Some extentions of the AdS/CFT correspondence
Outline Introduction to higher derivative gravities Higher derivative gravity and AdS/CFT Lovelock gravities Interacting fermions and AdS/CFT Implications for field theories Comments We have seen that adding higher curvature terms to the usual AdS/CFT setup gives useful insight for the physics of CFTs. Other directions include ◮ Applications in non-relativistic holography ◮ Tests for supersymmetrizability of higher derivative gravities (HDG) ◮ Black hole solutions with novel properties Andrei Parnachev Some extentions of the AdS/CFT correspondence
Recommend
More recommend
