APD-Invariant Tensor Networks from Matrix Quantum Mechanics

We propose a simple connection between matrix quantum mechanics and tensor networks. This allows us to imbue tensor networks with some interesting additional structure. The geometry of the graph describing the tensor network state is determined dynamically, giving a notion of background independence. The tensor network states have a $U(N)$ invariance, which (a) allows us to consider continuous families of entanglement cuts even with a finite number of tensors and (b) includes a notion of bulk coordinate reparameterization and area-preserving diffeomorphism invariance in the large N limit. These tensor networks also have a natural scale of nonlocality that behaves similarly to a string scale, suggesting a potential toy model for sub-AdS physics. Emergent $U(p)$ gauge fields naturally appear on the tensor network links.