Entwinement and the emergence of spacetime

TL;DR

This work introduces entwinement, a gauge-aware analogue of entanglement for internal degrees of freedom, to access regions of spacetime not probed by spatial entanglement. In AdS3/CFT2 with a conical defect, spatial entanglement via the RT prescription reconstructs geometry only down to a finite radius, creating an entanglement shadow; entwinement, defined via the covering space and summing over images, corresponds to non-minimal geodesics and recovers the interior up to the defect. The authors develop a reconstruction framework showing bulk curves can be determined from boundary entanglement together with entwinement, illustrating that fractionated, gauged degrees of freedom govern interior emergence. They discuss extensions to BTZ and higher dimensions, the role of mutual information, and implications for black hole interiors and holographic reconstruction beyond simple defects.

Abstract

It is conventional to study the entanglement between spatial regions of a quantum field theory. However, in some systems entanglement can be dominated by "internal", possibly gauged, degrees of freedom that are not spatially organized, and that can give rise to gaps smaller than the inverse size of the system. In a holographic context, such small gaps are associated to the appearance of horizons and singularities in the dual spacetime. Here, we propose a concept of entwinement, which is intended to capture this fine structure of the wavefunction. Holographically, entwinement probes the entanglement shadow -- the region of spacetime not probed by the minimal surfaces that compute spatial entanglement in the dual field theory. We consider the simplest example of this scenario -- a 2d conformal field theory (CFT) that is dual to a conical defect in AdS3 space. Following our previous work, we show that spatial entanglement in the CFT reproduces spacetime geometry up to a finite distance from the conical defect. We then show that the interior geometry up to the defect can be reconstructed from entwinement that is sensitive to the discretely gauged, fractionated degrees of freedom of the CFT. Entwinement in the CFT is related to non-minimal geodesics in the conical defect geometry, suggesting a potential quantum information theoretic meaning for these objects in a holographic context. These results may be relevant for the reconstruction of black hole interiors from a dual field theory.

Figures (5)

• Figure 1: A conical defect geometry as a wedge cut out of AdS space.
• Figure 2: A spatial slice of anti-de Sitter space (left) and of the conical defect ${\rm AdS }_3/ \mathbb{Z}_n$ (right). Spatial geodesics in the conical defect geometry descend from geodesics in the covering space with one endpoint ranging over the $n$ images. All but one of them are long geodesics.
• Figure 3: The holographic computation of the entanglement entropy of an interval (green) of width $2\alpha < \pi$ (above) and $2\alpha > \pi$ (below), shown in the short string picture (left) and in the long string picture (right). The short string interval maps to a union of disjoint long string intervals. The geometries on the left represent the spatial slice of the conical defect while the geometries on the right are spatial slices of anti-de Sitter space, which is the $n$-fold cover of the conical defect. There is a transition in the shortest geodesic homologous to the boundary interval when $2\alpha = \pi$ . Here $n=5$.
• Figure 4: A strand of three target space fields $X^{1,2,3}$, which define a single field $\tilde{X}^1$ of the long string.
• Figure 5: The integrand of eq. (\ref{['adscurve0']}) as a finite difference: $\frac{1}{2}\frac{d\,l(\alpha)}{d\alpha}\,d\theta \approx l(\alpha) - l(\alpha-d\theta/2) = {\color{blue} l(\alpha)} - \frac{1}{2}\, {\color{red} l(\alpha-d\theta/2)} - \frac{1}{2}\, {\color{red} l(\alpha-d\theta/2)} \approx \frac{R_0 \,d\theta}{4G}$The color-coded summands correspond to the continuously drawn pieces of the geodesics in the figure. The difference between their lengths aligns with the length element along the central circle $R = R_0$.