From path integrals to tensor networks for the AdS/CFT correspondence

In this paper, we discuss tensor network descriptions of AdS/CFT from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti-de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS(3) in terms of the dual two-dimensional conformal field theory (CFT). Both approaches support a correspondence between the hyperbolic time slice H-2 in AdS(3) and a version of continuous multiscale entanglement renormalization ansatz. We also give a heuristic argument about why we can expect a sub-AdS scale bulk locality for holographic CFTs.