Higher Curvature Gravity from Entanglement in Conformal Field Theories

121

172

We study the boundary description of the volume of maximal Cauchy slices using the recently derived equivalence between bulk and boundary symplectic forms. The volume of constant mean curvature slices is known to be canonically conjugate to "York time". We use this to construct the boundary deformation that is conjugate to the volume in a handful of examples, such as empty AdS, a backreacting scalar condensate, or the thermofield double at infinite time. We propose a possible natural boundary interpretation for this deformation and use it to motivate a concrete version of the complexity=volume conjecture, where the boundary complexity is defined as the energy of geodesics in the Kähler geometry of half sided sources. We check this conjecture for Bañados geometries and a mini-superspace version of the thermofield double state. Finally, we show that the precise dual of the quantum information metric for marginal scalars is given by a particularly simple symplectic flux, instead of the volume as previously conjectured.Here, we are denoting linearized deformations by |δΨ λ ≈ |Ψ λ+δλ − |Ψ λ , the bulk field φ is dual to the operator sourced by λ, its conjugate momentum is π, and Σ is a bulk initial value surface. This is the main result of [21] and it gives a

Section: Example: Modular Flowsupporting

confidence: 63%

Complexity and the bulk volume, a new York time story

Belin

Lewkowycz

Sárosi

2019

121

172

“…A multi-trace insertion involving k operators is denoted by the k operators being connected by dashed lines with k ends. 15 For example:…”

Section: Nonlocal Multi-trace Deformationsmentioning

confidence: 99%

“…We will assume that for the one-parameter family of CFT states |Ψ(λ) , we have found a one-parameter family of states of a corresponding gravitational theory, described by asymptotically AdS geometry M (λ) and bulk quantum state |Ψ bulk (λ) for the fields on M (λ). We assume that the gravitational state correctly computes the CFT entanglement entropy via 37 It is instructive to remind the reader of the derivation of classical second order Einstein equations in [14,15]: there, the classical analog of (7.20) was used, where all objects are truncated at order N 2 . At the classical level, one clearly has ST = HW = KH = 0.…”

Section: Quantum-gravitational Constraintsmentioning

confidence: 99%

Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories

Haehl

Mintun

Pollack

et al. 2019

Self Cite

We consider CFT states defined by adding nonlocal multi-trace sources to the Euclidean path integral defining the vacuum state. For holographic theories, we argue that these states correspond to states in the gravitational theory with a good semiclassical description but with a more general structure of bulk entanglement than states defined from single-trace sources. We show that at leading order in large N , the entanglement entropies for any such state are precisely the same as those of another state defined by appropriate single-trace effective sources; thus, if the leading order entanglement entropies are geometrical for the single-trace states of a CFT, they are geometrical for all the multi-trace states as well. Next, we consider the perturbative calculation of 1/N corrections to the CFT entanglement entropies, demonstrating that these show qualitatively different features, including non-analyticity in the sources and/or divergences in the naive perturbative expansion. These features are consistent with the expectation that the 1/N corrections include contributions from bulk entanglement on the gravity side. Finally, we investigate the dynamical constraints on the bulk geometry and the quantum state of the bulk fields which must be satisfied so that the entropies can be reproduced via the quantum-corrected Ryu-Takayanagi formula. arXiv:1904.01584v2 [hep-th] 5 Jun 2019 7 CFT entanglement entropy and the quantum RT formula 42 7.1 Basic identities 42 7.2 Quantum-gravitational constraints 49 7.2.1 Direct argument for the equality of relative entropies 50 7.2.2 Quantum-corrected Einstein equation: first order in the graviton operator 53 7.2.3 Second order "quantum" Einstein equations 56 8 Summary and discussion 57 -i -A General correlation functions in the effective geometry 59 B Divergent relative entropy for perturbed CFT thermal states 62 B.1 Relative entropy in thermal states using replica trick 62 B.2 Relative entropy in thermal states with finite spatial separation 66 C Perturbative relative entropy at k-th order 67 1 Here, N is a parameter related to the number of degrees of freedom in the theory; for example, the rank of a gauge group, or some power of the central charge in a CFT. 2 In [13] this argument was generalized to linearized equations about more general background states. 3The second-order calculation is sensitive to two central charge parameters in the CFT, characterizing respectively the vacuum entanglement of balls and the stress tensor two-point function. If these a-and c-type central charges are equal, the gravitational equations are those of pure Einstein gravity, otherwise the auxiliary gravitational theory involves higher derivatives [15,16].5 Similar observations regarding the breakdown of naive perturbation theory have been made recently in [23,24]. 6 See also [28][29][30][31] for similar recent constructions. 10 We thank O. Parrikar for raising this point.

“…In this paper, we derived the linearized equation of motion of bulk gravity in the presence of CS term. For AdS/CFT with Einstein gravity and higher derivative gravity, the equation of motion beyond linearized order were fulfilled in [9,10]. It is interesting and important to see whether some ideas and methods there can be extended to incorporate the CS term and gravitational anomalies.…”

Section: Equation Of Motion Beyond the Linearized Ordermentioning

confidence: 99%

“…More specifically, in the framework of AdS/CFT correspondence, the authors in [7] derived the linearized Einstein equation in AdS from entanglement. The generalizations to higher derivative gravities and Einstein equation at the non-linear level were also considered later in [8][9][10]. This provides a promising and direct way to help understand how the bulk gravity is assembled in the field theory.…”

Section: Introductionmentioning

confidence: 99%

Anomalous gravitation and its positivity from entanglement

Jiang

2019

We explore the emergence of gravitation from entanglement in holographic CFTs with gravitational anomalies. More specifically, the holographic correspondence between topologically massive gravity (TMG) with gravitational Chern-Simons term in the 3D bulk and its dual CFT with unbalanced left and right moving central charges on the 2D boundary, is studied from the quantum entanglement perspective. Using the first law of entanglement, we derive the holographic dictionary of the energy-momentum tensor in TMG, including the chiral case with logarithmic mode. Furthermore, we show that the linearized equation of motion of TMG can also be obtained from entanglement using the Wald-Tachikawa covariant phase space formalism. Finally, we identify a quasi-local gravitational energy in the entanglement wedge as the holographic dual of relative entropy in gravitationally anomalous CFTs. The positivity and monotonicity of relative entropy imply that such a gravitational energy should be positive definite and become larger when increasing the size of the entanglement wedge. These constraints from quantum information may be potentially used to discuss the UV inconsistent issues of TMG.