Entanglement, Holography and Causal Diamonds

TL;DR

This work reorganizes CFT data through observables defined on the moduli space of causal diamonds, a 2$d$-dimensional manifold with metric signature $(d,d)$. It introduces $Q(\\mathcal{O};x,y)$, a nonlocal observable built by smearing the one-point function of a primary operator over causal diamonds, and shows it obeys simple linear (and in 2d, nonlinear) equations of motion on the moduli space, linking to the OPE and the first law of entanglement. In holographic CFTs, $Q$ has a bulk dual given by Integrals of dual fields over RT surfaces; for 2d CFTs, the dynamics on the kinematic space factorizes into two de Sitter factors and extends to nonlinear Liouville/Toda-type equations for entanglement and higher-spin entanglement entropies. The paper also develops higher-spin generalizations via Wilson loops in Chern-Simons theory, demonstrating spin-two and spin-three entanglement entropies that satisfy interacting dS field equations and discusses the challenges of extending nonlinear dynamics beyond the linear regime. Overall, the framework offers a novel geometric and holographic lens on CFT degrees of freedom, entanglement structure, and potential dynamical theories living on the space of causal diamonds.

Abstract

We argue that the degrees of freedom in a d-dimensional CFT can be re-organized 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 2d-dimensional 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.

Figures (12)

• Figure 1: A causal diamond (in $d=3$ dimensions) and our basic coordinates. Specifying the timelike separated pair of points $(x^\mu,y^\mu)$ is equivalent to specifying a spacelike $(d-2)$-sphere which consists of all points $w^\mu$ null separated from both $x^\mu$ and $y^\mu$, i.e., satisfying Eq. (\ref{['eq.spherecond']}). The alternative parametrization in terms of $c^\mu = \frac{1}{2}(y^\mu+x^\mu)$ and $\ell^\mu = \frac{1}{2} (y^\mu-x^\mu)$ will prove convenient in section \ref{['sec:CausalDi']}.
• Figure 2: Anti-de Sitter hyperboloid in flat embedding space $\mathbb{R}^{2,d}$ is indicated in blue. The timelike embedding coordinates are $X^{-}$ and $X^0$. The remaining directions (including the $d-1$ suppressed dimensions $X^{2,\,\cdots,d}$ at each point) are spacelike. The green $d$-plane is orthogonal to the timelike vector $T^b$ and to the spacelike vector $S^b$ (the latter being hidden in the suppressed dimensions). The intersection of the $d$-plane with AdS$_{d+1}$ yields the green minimal surface. Its boundary as the hyperboloid approaches the red lightcone defines a $(d-1)$-sphere in the CFT.
• Figure 3: Lightcone coordinates for two-dimensional causal diamonds. The coordinates $(\xi,\bar{\xi})$ will provide a useful parametrization of the given diamond in section \ref{['sec.genO']}. Changing the endpoints corresponds to moving in the moduli space of causal diamonds parametrized by $(u,\bar{u},v,\bar{v})$; thereby $u$ is constant if $x^\mu$ moves along the line $\xi=u$, and so forth.
• Figure 4: Three basic types of infinitesimal moves in the moduli space of causal diamonds (in $d=3$ dimensions): (a) spacelike moves correspond to translations of the diamond, (b) timelike moves correspond to deformations of the diamond which leaves its centre fixed, (c) null moves correspond to a combination of the previous two by the 'same' amounts.
• Figure 5: Illustration of lightcone singularities in the moduli space of causal diamonds. We compare the big blue reference diamond $\lozenge_1$ with the small blue diamond $\lozenge_2$. If the red (green) tip of $\lozenge_2$ leaves the red (green) shaded lightcone region, the geodesic distance $\mathbf{d}(\lozenge_1,\lozenge_2)$ becomes infinite, i.e., the diamonds are no longer geodesically connected. An example of this happening would be by moving the tip $x_2$ along the arrow towards the lightcone of $y_1$.