Segmented strings and holography

In this paper we establish a connection between segmented strings propagating in ${\mathrm{AdS}}_{d+1}$ and ${\mathrm{CFT}}_d$ subsystems in Minkowski spacetime characterized by quantum information theoretic quantities calculated for the vacuum state. We show that the area of the world sheet of a string segment on the AdS side can be connected to fidelity susceptibility (the real part of the quantum geometric tensor) on the CFT side. This quantity has another interpretation as the computational complexity for infinitesimally separated states corresponding to causal diamonds that are displaced in a spacelike manner according to the metric of kinematic space. These displaced causal diamonds encode information for a unique reconstruction of the string world sheet segments in a holographic manner. Dually the bulk segments are representing causally ordered sets of consecutive boundary events in boosted inertial frames or in noninertial ones proceeding with constant acceleration. For the special case of ${\mathrm{AdS}}_3$ one can also see the segmented stringy area in units of $4GL$ ($G$ is Newton’s constant and $L$ is the AdS length) as the conditional mutual information $I(A,C|B)$ calculated for a trapezoid configuration arising from boosted spacelike intervals $A$, $B$ and $C$. In this special case the variation of the discretized Nambu-Goto action leads to an equation for entanglement entropies in the boundary theory of the form of a Toda equation. For arbitrary $d$ the string world sheet patches are living in the modular slices of the entanglement wedge. They seem to provide some sort of tomography of the entanglement wedge where the patches are linked together by the interpolation ansatz, i.e., the discretized version of the equations of motion for the Nambu-Goto action.

Figure 1

The spacelike minimal surface $X_{\mathcal{R}}$, a spacelike geodesic (red), defined by the intersection of the cones $U·X=0,V·X=0$ in the Poincaré patch. In the figure the $x_u=x_v=x_0=0$, $t=t_0=\frac{1}{2}(t_u+t_v)$ case is shown. On the boundary (the gray plane with $z=0$) the cones give rise to a causal diamond (black) of the blue region $\mathcal{R}$. Note that both the blue region and the red minimal surface is on a $t=t_0=\text{const}$ hypersurface (Cauchy slice) given by first equation in (19).

Figure 2

The static subregion $R$ (blue) with its causal diamond. For the extremal surface homologous to it see Fig. 1.

Figure 3

A static world sheet of a string segment (green rectangle) in the $x=0$ plane of the Poincaré patch. The vertices of the segment are defined by the Poincaré representatives $V_i^{\widehat{\mu }}$ of the vectors $V_i\in {\mathrm{AdS}}_3\subset {\mathbb{R}}^{2,2}$ represented by black dots with the respective numbering. On the other hand the edges are labeled by the null vectors $p_1$, $p_2$, $p_3$, $p_4$. The edges define light rays (dashed lines) intersecting the boundary (for the static configuration the $t$ axis) in the points $t_1$, $t_2$, $t_3$, $t_4$.

Figure 4

The correspondence between segmented strings, extremal (minimal) surfaces, and boundary subregions. The world sheet of a string segment as parametrized in Fig. 3 is shown in green. The edges of the world sheet are parametrized by null vectors. The neighboring null vectors give rise to extremal (minimal) surfaces (red). They in turn determine causal diamonds of subregions (blue) in the boundary. In the figure the special case where the boundary regions are parallel is shown. Here the tips of the causal diamonds are having varying $t$ coordinates but all have $x=0$, hence they are lying on a time axis of an inertial system.

Figure 5

Causally ordered sets of boundary points representing consecutive events occurring in inertial, boosted inertial and noninertial frames (frames with constant acceleration) exhibiting hyperbolic motion. According to Eq. (80) the acceleration, corresponding to the situation represented by the rightmost figures, is connected to the normal vector of the world sheet (shown in green) of the string segment. In the upper part of these figures the bulk perspective displaying the lightlike geodesics needed for the reconstruction of the sheet in a holographic manner is also shown. For the corresponding causal diamonds of the leftmost figure see Fig. 4.

Figure 6

The boundary perspective of the string world sheet and the boundary regions hidden by minimal surfaces (red line segments) as seen from the bulk direction. The green line segment is the projection of the string world sheet segment. The red lines are the minimal surfaces homologous to the boundary regions; ${\mathcal{R}}_{12}$, ${\mathcal{R}}_{34}$, ${\mathcal{R}}_{23}$, ${\mathcal{R}}_{14}$. The corresponding causal diamonds (black) are also shown. For a 3D perspective see Fig. 4.

Figure 7

The parametrization of the vertices of the causal diamonds living on the boundary. It is the $z=0$ section of Fig. 4. The coordinates are light cone ones.

Figure 8

The arrangement of causal diamonds of the boundary needed for establishing a relation between the area of the world sheet of a string segment and conditional mutual information of boundary subregions. The diamonds have the same intersection and union followed by causal completion as the ones of Fig. 7. The four blue lines in this case contain two diagonal ones corresponding to the new regions ${\mathcal{R}}_{d_1}$ and ${\mathcal{R}}_{d_2}$. They are featuring the trapezoid configuration of Fig. 9 and Eq. (144).

Figure 9

The trapezoid configuration living inside of Fig. 8. The top ($B$) and bottom segments ($ABC$) are the ones ${\mathcal{R}}_{34}$ and ${\mathcal{R}}_{12}$ of Fig. 6. However, the remaining segments ${\mathcal{R}}_{23}$ and ${\mathcal{R}}_{14}$ from Fig. 6 are replaced by ${\mathcal{R}}_{d_1}$ and ${\mathcal{R}}_{d_2}$ representing the diagonals $AB$ and $BC$ of the trapezoid.

Figure 10

An illustration of the connection between the discretized variables of segmented strings to the ones showing up in the continuous one [20]. In the notation of Sec. 3a clearly we have $V_1\leftrightarrow X$, and ${\partial }_{-}X\leftrightarrow p_1$, ${\partial }_{+}X\leftrightarrow p_4$. The equations in red define the cones with future ($y$) and past ($x$) tips. Their intersections give rise to the extremal surface (red line) which is now a spacelike geodesic. This figure should be compared to Fig. 1.

Figure 11

A world sheet built up from string segments. Each side of a given segment is labeled by its defining lightlike vector $p_{ij},p_{i+1j},p_{i+1,j-1}$ or $p_{ij-1}$.