Relationship between Bulk and Boundary Operators

As part of the holographic dictionary operators in the bulk also have a correspondence to operators on the boundary. While there is a correspondence, it is not unique and a bulk operator at a point is equivalent to multiple non-local boundary operators, possibly over different regions. The opposite direction also holds and a boundary operator may also be expressed as a limit of boundary operators.

On the boundary we define operators through primary fields O(x), where x is the bound-ary coordinates. If we consider the boundbound-ary field with scaling dimension ∆ then the boundary operator may be expressed in terms of limits of bulk operator φ(x, z). The bulk operator is a field on the bulk manifold (generally some time slice of asymptotically AdS space) and for example could be a scalar field, a gauge field, or any other type of field.

This realisation of the correspondence is called the extrapolation dictionary and states that O(x) ≡ lim_z→∞z^∆φ(x, z) where z is the coordinate of the additional dimension in the bulk [48]. Again, this additional coordinate can also be interpreted as the renormal-isation direction of the boundary field.

The reverse procedure is also possible and called the HKLL procedure[49]. In this pro-cedure the bulk operator φ(x, z) can be expressed as a CFT operator on the boundary with non-trivial support only on some spatial subregion A of a boundary time slice.

However this is only possible if (x, z) corresponds to a point in the entanglement wedge of the region A, denoted W_E[A] (defined later in this subsection). In which case it is possible to write φ(x, z) =R

AK(x⁰|x, z)O(x⁰)dx⁰ where K(x⁰|x, z) is called the smearing function which connect bulk operators to boundary operators based on the bulk geometry.

The smearing functions are not guaranteed to be unique, nor to exist, though they have been computed for a number of AdS geometries [49, 50]. The existence and uniqueness of a smearing function is not of importance in this thesis and so these questions will not be explored here. However in section 5.5 something similar to the smearing function is constructed, based on the tensors which construct the MERA.

The original paper which gave rise to the HKLL procedure [49] was actually restricted to the bulk operator φ(x, z) being in the causal wedge rather then the entanglement wedge.

Later work by Dong, Harlow and Wall extended this to the entanglement wedge [48].

This was partially motivated by a reinterpretation of the HKLL procedure in terms of quantum error correction [50], connecting this bulk boundary correspondence to quantum information ideas.

Both causal wedges [49] and entanglement wedges [51, 52, 53] are defined only for bulk space-time manifolds M with an asymptotic boundary space-time manifold B. Each wedge is defined for some sub-region A of a time slice on the boundary Σ ⊂ B. This region has a causal diamond D which is the region of support of A on the boundary (see figure 3.3a). For point p in the causal diamond D this means that the collection of points on Σ the which are time-like connected to p are all in region A. The causal wedge is then defined to be the set of points in the bulk which can both send and receive signals from the causal diamond (but may send/receive signals from other points on the boundary as well).

On the other hand the entanglement wedge given some time slice can be defined in a much simpler fashion. Given the boundary A the extremal surface γ_A is a codimension one surface in the bulk with a boundary matching that of region A (see figure 3.3b). In the case there is not a single unique surface the minimal surface is chosen. The time slice of the entanglement wedge, R ⊂ WE[A], is then all the points that can be reached from any point in region A without crossing the minimal surface γ_A. This definition is satisfactory for bulk time slices ˜Σ, however to match up with the causal wedge picture from figure 3.3a and define it over the full spacetime the entanglement wedge WE[A] is defined as the domain of dependence of R. This is the set of points that are only time-like connected to a subregion R⁰ ⊂ R on the ˜Σ time-slice. Therefore this is the collection of point that can be completely defined by region R in ˜Σ. Further the entanglement wedge includes the causal wedge in cases of regular interest[54], satisfying certain reasonable assumptions which I will not go into in this thesis

The boundary of the entanglement wedge on time slice ˜Σ in the bulk corresponds to the minimal surface of the Ryu-Takayanagi formula discussed in subsection 3.1.1. This pro-cedure can be extended to the case of disjoint boundaries where the entanglement wedge on the time slice ˜Σ is the shaded region in figure 3.2a. The region for the full space-time can be computed by working out the domain of dependence of the shaded region.

Something analogous to the HKLL formula is discussed in section 5.5 with respect to the lifting procedure developed by myself and collaborators. This procedure proposes a

con-a) b)

Figure 3.3: a) An example of an entanglement wedge for region A on a boundary of the time slice ˜Σ. This time slice is a hyperbolic space based represented by the Poincar´e disc, upon which is the minimal surface, γ_A, bounds the time slice of the entanglement wedge R ⊂ WE[A], marked in blue, as discussed in section 3.1.1. The causal diamond defined on the boundary and used to define the causal wedge is also indicated in this diagram by the red highlighting. The inclusion of both the causal diamond and the entanglement wedge on the time slice on the same diagram is simply for visualisation purposes as they are not related to each other. The causal diamond is important for the causal wedge and dictates which sites are within the wedge (the sites which can both send and receive a signal from the causal diamond). The entanglement wedge on the other hand is based off the time slice region R, the wedge being the domain of dependence of R. The entanglement wedge exists within the bulk and isn’t pictured on the figure but can be roughly visualised as having a light like boundary going from the minimal surface γAto the boundary of causal diamond D (or more accurately an analogous surface larger then D). b) Examples of two different regions A and B on the boundary of time slice ˜Σ.

These regions could encode a local bulk operator φ existing in the entanglement wedges (restricted to the time slice ˜Σ) of A and B respectively. The bulk operator in this figure indicates the holographic mapping between does not map each local bulk operator to a unique non-local boundary operator. Rather it can map them to multiple equivalent boundary operators over different regions on the boundary.

nection between the MERA and AdS/CFT correspondence, giving rise to a connection between operators defined on the MERA, i.e. on the boundary, and those defined in the bulk, an analogue to AdS space.