From Path Integrals to Tensor Networks for AdS/CFT

TL;DR

The paper links AdS/CFT to tensor-network descriptions via two routes: (i) an optimized Euclidean path-integral with a position-dependent UV cutoff that reproduces a hyperbolic time slice and (ii) a Lorentzian, surface/state–based flow that yields a continuous tensor network, essentially a cMERA, for the AdS$_3$/CFT$_2$ vacuum. Both approaches converge on a duality between the AdS$_3$ hyperbolic slice $H_2$ and a continuous tensor network, with de Sitter variants emerging for certain slicings and excited/bulk states informing connections to perfect/random tensor networks. A central argument shows that large-$c$ holographic CFTs admit sub-AdS locality through a refined, long-string–driven discretization, and the framework naturally extends to bulk-local excitations by tensor modifications, linking to Ishibashi/cross-cap structures and scrambled entanglement patterns. Collectively, the work advances a coherent picture in which cMERA-like continuous tensor networks capture AdS$_3$/CFT$_2$ geometry and dynamics, including sub-AdS locality and bulk excitations, while suggesting fruitful generalizations to higher dimensions and more complex backgrounds.

Abstract

In this paper, we discuss tensor network descriptions of AdS/CFT from two different viewpoints. First, we start with an Euclidean path-integral computation of ground state wave functions with a UV cut off. We consider its efficient optimization by making its UV cut off position dependent and define a quantum state at each length scale. We conjecture that this path-integral corresponds to a time slice of AdS. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS3 in terms of the dual 2d CFT. Both approaches support a correspondence between the hyperbolic time slice H2 in AdS3 and a version of continuous MERA (cMERA). We also give a heuristic argument why we can expect a sub-AdS scale bulk locality for holographic CFTs.

Figures (5)

• Figure 1: A computation of ground state wave function from Euclidean path-integral and its optimization, which is described by a hyperbolic geometry.
• Figure 2: Folding the MERA network in a 2d symmetric product CFT $M^m/S_m$ (we chose $m=4$.). The left picture expresses a MERA network for a long string sector vacuum which is equivalent to the single string sector vacuum with the radius $mR_0=4R_0$. The right picture describes its equivalent network after the folding such that the radius is $R_0$. We show the coarse-graining (isometries) as tri-vertices and the disentanglers (unitary transformations) as horizontal lines. The right network shows that the actual lattice spacing is $\epsilon/m$. From this network, we can easily see that the entanglement entropy $S_A$ follows the volume law for a small interval $A$ with the width $(\epsilon/m\ll) \Delta x\ll \epsilon$ in the large $c$ limit $m\to \infty$. Note also that the final MERA network is squeezed near the top region (roughly more than $\log {R_0\over \epsilon}$ steps from the bottom). This is very analogous to the global AdS$_3$ metric.
• Figure 3: A modification of tensor network dual to a bulk local excitation.
• Figure 4: The tensor network evolutions which corresponds to a locally excited state in global AdS$_3$.
• Figure 5: Gluing two cross cap states lead to an identity operation.