Random tensor networks (RTNs) have proved to be fruitful tools for modelling

the AdS/CFT correspondence. Due to their flat entanglement spectra, when

discussing a given boundary region $R$ and its complement $bar R$, standard

RTNs are most analogous to fixed-area states of the bulk quantum gravity

theory, in which quantum fluctuations have been suppressed for the area of the

corresponding HRT surface. However, such RTNs have flat entanglement spectra

for all choices of $R, bar R,$ while quantum fluctuations of multiple

HRT-areas can be suppressed only when the corresponding HRT-area operators

mutually commute. We probe the severity of such obstructions in pure AdS$_3$

Einstein-Hilbert gravity by constructing networks whose links are codimension-2

extremal-surfaces and by explicitly computing semiclassical commutators of the

associated link-areas. Since $d=3,$ codimension-2 extremal-surfaces are

geodesics, and codimension-2 `areas’ are lengths. We find a simple 4-link

network defined by an HRT surface and a Chen-Dong-Lewkowycz-Qi constrained HRT

surface for which all link-areas commute. However, the algebra generated by the

link-areas of more general networks tends to be non-Abelian. One such

non-Abelian example is associated with entanglement-wedge cross sections and

may be of more general interest.

Random tensor networks (RTNs) have been valuable in modeling the AdS/CFT correspondence. However, while standard RTNs have flat entanglement spectra for all choices of boundary regions, quantum fluctuations of multiple HRT-areas can only be suppressed if the corresponding HRT-area operators mutually commute. This poses a challenge in constructing networks using extremal-surfaces as links.

## Future Roadmap

### 1. Exploring Pure AdS$_3$ Einstein-Hilbert Gravity

A potential challenge in pure AdS$_3$ Einstein-Hilbert gravity is understanding the severity of obstructions caused by non-commuting link-areas in network construction. By constructing networks using codimension-2 extremal-surfaces as links and calculating the semiclassical commutators of the associated link-areas, we can investigate this problem further.

### 2. Finding Commuting Link-Areas

In the case of codimension-2 `areas’ being lengths and geodesics in $d=3,$ we discovered a simple 4-link network consisting of an HRT surface and a constrained HRT surface that commute. This finding suggests that it may be possible to identify other specific network configurations where all link-areas commute. This should be explored further to understand the limitations and opportunities associated with such networks.

### 3. Non-Abelian Algebra and Entanglement-Wedge Cross Sections

While the 4-link network provided a commutative algebra for link-areas, more general networks tend to have non-Abelian algebras. One example of a non-Abelian network is associated with entanglement-wedge cross sections. Investigating these non-Abelian networks and their properties is of interest for a deeper understanding of the AdS/CFT correspondence.

## Challenges and Opportunities

**Challenge:**The main challenge lies in understanding the severity of obstructions caused by non-commuting link-areas in network construction.**Opportunity:**The discovery of a 4-link network with commuting link-areas suggests that it may be possible to identify other specific configurations for which all link-areas commute.**Opportunity:**Exploring non-Abelian networks, such as the one associated with entanglement-wedge cross sections, can provide valuable insights into the AdS/CFT correspondence.

Key Takeaway:The study of random tensor networks in the context of the AdS/CFT correspondence has shown that while standard RTNs have flat entanglement spectra, non-commuting link-areas pose challenges in network construction. However, exploring specific network configurations and non-Abelian networks can provide opportunities for further understanding and advancements in this area.