Entanglement, holography and causal diamonds

Abstract

We argue that the degrees of freedom in a d-dimensional CFT can be reorganized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2ddimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglement entropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed. Contents 65 A.3 Moduli space of spacelike separated pairs of points 67 -i -B Conventions for symmetry generators 71 B.1 General definitions 71 B.2 Two-dimensional case 73 C Relative normalization of CFT and bulk quantities 74 C.1 Holographic computation for a free scalar in AdS 3 75Due to these considerations, in what follows we will interchangeably use the notions of spheres, causal diamonds, and pairs of timelike separated points.3 Implicitly, then we are assigning an orientation to the spheres, i.e., the interior is distinguished from the exterior. One could also consider unoriented spheres, which would amount to an additional Z2 identification in the coset given in Eq. (2.11). See [35] for further discussion. 4 Our notation here and throughout the following is that for d-dimensional vectors, (y − x) 2 = ηµν (y − x) µ (y − x) ν .

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

“…Curiously, from the general solution of the Liouville equation, we can now see that this result itself can also be written as a Liouville field and satisfies the Liouville equation but with positive curvature [76] and the space described by the end-points of the interval. It appears that these two Liouville fields can obtained form each other by simple analytic continuation (see also [56]) but the physical significance of this fact is far from obvious and remains to be elucidated.…”

“…In general we can use the first law for entanglement entropy in states conformally mapped to the vacuum (see e.g. [75,76]) and show that the increase in the entropy is proportional to the (constant) Liouville energy momentum tensor.…”

Space-time in the SYK model

“…There is another construction of the Radon transform which is used in [101][102][103]105] and is based on integration over geodesics. For the Euclidean AdS 2 space-time (3.2), a geodesic is given by a semicircle…”

“…It turns out that the corresponding position space kernel is a H 2 Radon transform. Radon transforms have appeared (explicitly or implicitly) in discussions of AdS/CFT, most notably in [101][102][103]105] where this is used to go from the bulk to the kinematic space of the boundary field theory on a time slice. Indeed the space on which the bi-locals live is a version of kinematic space.…”

On the Kerr-AdS/CFT correspondence

“…The purpose of the present work is to review the relation of semi-classical (global) conformal blocks in four dimensions to propagating (scalar) fields in five-dimensional anti-de Sitter (AdS), a fact outlined by [20] and [18,21,22] revisiting work by Ferrara and collaborators in the 70's [23][24][25]. The construction does not rely on supersymmetry.…”