  Holographic tensor networks model AdS/CFT, but so far they have been limited
by involving only systems that are very different from gravity. Unfortunately,
we cannot straightforwardly discretize gravity to incorporate it, because that
would break diffeomorphism invariance. In this note, we explore a resolution.
In low dimensions gravity can be written as a topological gauge theory, which
can be discretized without breaking gauge-invariance. However, new problems
arise. Foremost, we now need a qualitatively new kind of "area operator," which
has no relation to the number of links along the cut and is instead
topological. Secondly, the inclusion of matter becomes trickier. We
successfully construct a tensor network both including matter and with this new
type of area. Notably, while this area is still related to the entanglement in
"edge mode" degrees of freedom, the edge modes are no longer bipartite
entangled pairs. Instead they are highly multipartite. Along the way, we
calculate the entropy of novel subalgebras in a particular topological gauge
theory. We also show that the multipartite nature of the edge modes gives rise
to non-commuting area operators, a property that other tensor networks do not
exhibit.
