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From Path Integrals to Tensor Networks for AdS/CFT

Masamichi Miyaji, Tadashi Takayanagi, Kento Watanabe

arXiv:1609.04645v2hep-thquant-ph

TL;DR

The paper investigates how tensor networks may describe AdS/CFT, where the holographic mechanism remains unclear. It develops position-dependent-cutoff path integrals and AdS3 symmetry-based state flows; both support a cMERA-like description of the hyperbolic AdS3 slice, alongside a heuristic argument for sub-AdS locality.

  • Problem

    The mechanism underlying AdS/CFT remains mysterious, and tensor networks are a possible route to understanding its connection to quantum states.

    Source: §1 · p. 2

  • Method

    The paper studies Euclidean ground-state path integrals with position-dependent UV cutoffs and derives state flows from AdS3 Killing transformations expressed in the dual CFT.

    Source: §1 · p. 3, §3.1 · p. 7

  • Results

    The path-integral construction and AdS3 symmetry analysis support interpreting the hyperbolic slice as a cMERA-like network, and the paper gives a heuristic sub-AdS locality argument.

    Source: §8 · p. 27

  • Takeaways & Limitations

    The authors conclude that cMERA may be viewed as a continuous refinement of perfect and random tensor networks describing a canonical AdS time slice below the cutoff scale.

    Source: §8 · p. 27

    An explicit relation between the continuous-network properties and the spacetime metric, especially its time-like component, remains a future problem.

    Source: §8 · p. 27

Abstract

from arXiv · show

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.

1 Introduction

The introduction frames tensor networks as a possible way to clarify AdS/CFT and studies that connection through path integrals and AdS symmetries. The authors argue both approaches support relating an AdS3 hyperbolic slice to a cMERA-like network, while also considering excited states and sub-AdS locality.

1 Introduction

  • MERA was conjectured to represent AdS/CFT through real-space RG flow, while cMERA extends this idea to a continuum and manifests special conformal invariance [4] [10] [11] [12].The introduction notes that MERA has also been argued to correspond to de Sitter space or kinematical space [13] [15] [17].Source: §1 · p. 2
  • The surface/state correspondence associates codimension-two convex surfaces in Einstein-gravity spacetimes with dual-theory quantum states [18].Perfect and random tensor networks are presented as toy models, though the perfect-tensor state has a flat entanglement spectrum and is not a CFT vacuum [19, 20].Source: §1 · pp. 2–3
  • The paper uses position-dependent-cutoff Euclidean path integrals and AdS3 transformations in the surface/state correspondence to study tensor-network descriptions of AdS3/CFT2.It also examines locally excited states and gives a heuristic argument for sub-AdS-scale bulk locality.Source: §1 · p. 3
  • The paper focuses mainly on two-dimensional CFTs for simpler presentation, while noting that many results can generalize to higher-dimensional AdS/CFT.Its sections cover the path-integral construction, Killing-symmetry-based networks, sub-AdS locality, and locally excited states.Source: §1 · p. 3

2 AdS3 Geometry

This section reviews AdS3 geometry and its symmetries, including both hyperbolic and de Sitter slices. It identifies the SL(2, R) transformations that preserve the t = 0 hyperbolic slice.

2.1 Global Symmetry

  • AdS3 has SL(2, R)L × SL(2, R)R symmetry, generated by operators corresponding to global Virasoro symmetry in its dual 2d CFT.The SL(2, R) subgroup preserving t = 0 is generated by l_n = L_n − ˜L_−n for n = 0, ±1.Source: §2.1 · p. 4
  • The preserving SL(2, R) generators satisfy the algebra and correspond to Killing vectors of the hyperbolic space H2.The section relates their geometrical action to linear combinations of the generators and uses an SL(2, R) transformation to map the origin to a point on H2.Source: §2.1 · p. 5

2.2 Hyperbolic/De Sitter Slices in AdS3

  • AdS3 admits de Sitter slices as well as hyperbolic slices; the de Sitter slices are constant-η surfaces within −π < t < 0, with the η = 0 limit becoming a light cone.A separate coordinate description is introduced for the hyperbolic slices.Source: §2.2 · pp. 5–6

3 AdS/CFT from Euclidean Path-integral

The section develops optimized Euclidean path integrals as scale-dependent state constructions and interprets their geometries in AdS/CFT. It extends the discussion to thermal and excited states and relates Lorentzian AdS slices to cMERA-like networks.

3 AdS/CFT from Euclidean Path-integral

  • The section begins by proposing that a conformally invariant position-dependent cutoff deforms the ground-state path integral onto H2, interpreted as an AdS3 time slice.The construction is presented as an efficient way to introduce the UV cutoff.Source: §3 · p. 6, §3.1 · p. 7

3.1 Euclidean Path-integral with UV cut off

  • A position-dependent UV cutoff restricts the path integral to contributing modes |k| ≲ 1/z, approximately preserving the ground-state wave function while making the computation more efficient.At finite z0, the resulting state reproduces the ground state below cutoff and becomes trivial at much higher momenta, describing a scale-dependent state in a real-space RG flow.Source: §3.1 · pp. 7–8

3.2 Interpretations in terms of AdS/CFT

  • The optimized path-integral is assigned a hyperbolic metric whose area counts discretized sites, with its z-direction fixed by SL(2, R) invariance; in a large-c CFT, the site density reproduces the AdS3 metric up to an O(1) constant.The argument is restricted to large-c CFTs; for central charge O(1), the strict UV cutoff prevents reaching the volume-law phase.Source: §3.1 · p. 8, §3.2 · p. 10

3.3 Einstein-Rosen Bridge from Path-Integral

  • For a thermo-field double state, the path-integral metric coincides with the time slice of a BTZ black hole, or Einstein-Rosen bridge.The state is prepared by a Euclidean path integral on a finite-width cylinder.Source: §3.3 · p. 11

3.4 More General Backgrounds

  • With a mass gap, the region z ≫ 1/m can be removed from the optimized lattice, qualitatively matching time slices dual to confining gauge theories.For deconfined excited states, the lattice retains sites even at z = ∞, interpreted as a black-hole horizon; for time-dependent excited states, a definite gravity-dual connection is harder to establish because no criterion for choosing time slices is given.Source: §3.4 · p. 12

3.5 Lorentzian AdS/CFT

  • The time-independent excited states considered in Lorentzian AdS3 are expected to correspond to CFT tensor networks that can be approximated by MERA.Analytic continuation maps Euclidean AdS3 to either hyperbolic or de Sitter slices, and the free-scalar scale-dependent wave function is essentially the cMERA state.Source: §3.5 · pp. 12–13

4 cMERA

cMERA builds a scale-dependent state by adding entanglement while coarse-graining from an IR state toward the UV. The section develops this construction for a free scalar and on a circle, where scale transformations have a quantum-quench interpretation.

4.1 Construction of cMERA

  • cMERA starts from an IR state and uses a local entangler K(u) and coarse-graining L to add entanglement across scales; u runs from the IR limit to the UV.For compact spaces the IR state is disentangled, while for noncompact CFTs it is the vacuum below the UV cutoff unless a massive deformation is introduced.Source: §4.1 · pp. 13–14

4.2 Free massless scalar field theory

  • For a free massless scalar, the Gaussian entangler implements a Bogoliubov transformation from the IR disentangled state to the CFT vacuum.The scale-dependent momentum definition identifies the path-integral wave-function state with the cMERA state, supporting the correspondence between the Lorentzian AdS hyperbolic slice and cMERA.Source: §4.2 · p. 16

4.3 Compactification and Space-like Scale Transformation

  • The space-like scale transformation L is argued to act as a quantum quench that changes the spatial radius from R to R(1 + η).In the compactified construction, the radius depends on scale in the |Ψ(u)⟩ picture but remains fixed in the |Φ(u)⟩ picture.Source: §4.3 · p. 17
  • In the compactified scalar theory, the evolution from the CFT vacuum to the IR Ishibashi state is realized as a quantum quench induced by changing the radius.The IR state is identified with the vacuum-sector Ishibashi state.Source: §4.3 · p. 19

5 Continuous Tensor Networks and AdS3/CFT2

The paper constructs continuous tensor networks from AdS3/CFT2 surface/state correspondence by translating AdS Killing flows into CFT operations. The resulting network agrees with canonical cMERA below the UV cutoff, and vacuum states on hyperbolic and de Sitter slices have equivalent cMERA properties.

5 Continuous Tensor Networks and AdS3/CFT2

  • A foliation of AdS3 by closed curves assigns a CFT state to each curve, yielding a continuous tensor network for a codimension-one slice.The construction uses the surface/state correspondence, in which a convex closed curve corresponds to a state in the dual CFT Hilbert space.Source: §5 · p. 19

5.1 Constructing Continuous Tensor Network from AdS3/CFT2

  • The AdS3-derived continuous tensor network agrees with canonical cMERA below the UV cutoff, with the scale evolution identified through the CFT radius.The authors state that they do not know how to probe quantum states from AdS3 above the cutoff.Source: §5.1 · pp. 20–21

5.2 Other Slices

  • For the CFT vacuum, hyperbolic and de Sitter slices have equivalent cMERA properties, but excited states yield different continuous tensor-network descriptions of the same state.The de Sitter slice additionally has a nonzero minimum radius.Source: §5.2 · p. 21

6 An Argument for Sub-AdS Scale Locality

The section argues heuristically that symmetric-product CFTs can support bulk resolution below the AdS scale. In the long-string sector, the finer momentum quantization yields a network lattice spacing estimated at the Planck scale.

6 An Argument for Sub-AdS Scale Locality

  • For the general cMERA construction, the authors find locality only at the AdS-radius scale because distinct bulk points cannot be distinguished at ρ = O(1).This motivates using a large-central-charge symmetric-product CFT as a tractable model for finer resolution.Source: §6 · p. 22
  • The long-string sector gives a network with structure finer by 1/m = O(1/c), corresponding to an estimated resolution L/c at the Planck scale.The argument treats the long-string sector as a single CFT on a cylinder of extended radius mR0, with actual lattice spacing estimated as ǫ/m.Source: §6 · p. 22
  • The sub-AdS locality argument is heuristic in AdS3/CFT2; extending it to higher-dimensional AdS/CFT involves a fractional power of central charge.The authors note that this suggests a more complicated characterization of holographic CFTs in higher dimensions.Source: §6 · p. 22

7 Bulk Locally Excited States and cMERA

The section identifies the CFT dual of a bulk-local excitation and constructs a tensor-network description for it. On the hyperbolic slice, the network can encode the excitation through a single-tensor modification, while the infrared state and other slices introduce qualifications.

7 Bulk Locally Excited States and cMERA

  • A regulated global Ishibashi state provides the CFT description of the bulk-local excitation.The section states that the UV regularization makes the excited energy finite.Source: §7 · p. 24
  • The CFT dual of a bulk-local excitation is identical to the state obtained by acting with the known HKLL bulk-field map on the vacuum.The excitation is created by a bulk scalar field at a point in global AdS3 and labeled by a primary field in the two-dimensional CFT.Source: §7 · p. 24

7.1 Continuous Tensor Network for Bulk Locally Excited States

  • On the hyperbolic slice, a bulk-local excitation can be represented by modifying only the tensor at the insertion point.The surrounding network uses the same tensor operator as the vacuum construction.Source: §7.1 · p. 25
  • The hyperbolic-slice construction does not yet extend systematically to general slices, where tensors inside the excitation's light cone would need modification.For the hyperbolic slice, modifying a single tensor—or equivalently the IR state—suffices.Source: §7.1 · p. 25
  • As the radius shrinks in the IR, the excited state's network flow approaches a cross-cap state in holographic CFTs under a large-c approximation.In free-field CFTs, gluing two cross-cap states over more than half of each gives the identity, a property also associated with perfect tensors [19, 43].Source: §7.1 · pp. 25–26
  • The authors suggest that holographic-CFT cross-cap states may be more scrambled than the simple entangled states found in free-field CFTs.This suggestion draws on entanglement scrambling and compares the holographic states to random tensors [20].Source: §7.1 · p. 26

8 Conclusion and Discussion

The paper argues that two approaches connect AdS time slices with cMERA: a position-dependent-cutoff path integral and a network derived from Lorentzian AdS geometry. The conclusion extends this picture to excited states and possible de Sitter slices, while identifying open questions about locality and the metric.

8 Conclusion and Discussion

  • The position-dependent UV cutoff preserves the final wave function, and its path integral on hyperbolic space yields scale-dependent states essentially matching cMERA below the cutoff.The authors argue that this supports interpreting a hyperbolic AdS time slice as a tensor network.Source: §8 · pp. 26–27
  • The Lorentzian AdS3 construction gives a network for the canonical H2 slice that essentially coincides with compactified cMERA, for modes below the cutoff.This construction assumes the surface/state correspondence.Source: §8 · p. 27
  • Analysis of locally excited bulk states finds the network analogous to perfect and random tensor networks, while symmetric product CFTs motivate a heuristic argument for sub-AdS bulk locality.The authors suggest cMERA may be a continuous refinement of those networks for describing canonical AdS slices below the cutoff.Source: §8 · p. 27
  • The Killing-vector analysis also supports fitting cMERA to de Sitter slices, since noncanonical Euclidean hyperbolic slices can Wick rotate to hyperbolic or de Sitter slices.The authors note a possible connection to kinematic spaces.Source: §8 · p. 27
  • The discussion mainly treats 2d CFTs for simplicity, while stating that most results generalize straightforwardly to higher-dimensional AdS/CFT.Source: §8 · p. 27
  • An explicit relation between spacetime metric properties and continuous tensor-network properties remains a future problem, especially for the time-like metric component.Source: §8 · p. 27

A Other Choices of UV Cut Off Function

The appendix varies the position-dependent UV cutoff while retaining the low-momentum behavior, producing wave functions with distinct high-momentum limits. One choice recovers the original cMERA IR state, while another yields a Dirichlet boundary state and Wilsonian-like suppression.

A Other Choices of UV Cut Off Function

  • The cutoff functions are varied while preserving low-momentum behavior for |k|z < 1 and scaling symmetry, thereby changing the higher-energy part of the scale-dependent wave function.Source: A Other Choices of UV Cut Off Function · p. 28
  • The first cutoff choice gives a high-momentum behavior coinciding with the original cMERA IR state |ΩΛ⟩ after rescaling.The authors describe it as a refinement of their earlier wave function.Source: A Other Choices of UV Cut Off Function · pp. 28–29
  • For the second choice with β2 = 1/2, the low-momentum coefficient reproduces the vacuum as z0 approaches zero, while high momenta approach a Dirichlet Ishibashi boundary state.The large scalar-field action suppresses high-momentum modes, which the authors compare to Wilsonian renormalization-group flow.Source: A Other Choices of UV Cut Off Function · p. 29

B Spacetime Continuous Tensor Network

The spacetime continuous tensor network generalizes cMERA by assigning states and local entanglers to foliated surfaces in a gravitational spacetime. Its construction uses a flat-connection consistency condition to define a unitary network evolution.

B Spacetime Continuous Tensor Network

  • The continuous tensor network is proposed as a generalization of cMERA for non-AdS spacetimes, following the surface/state correspondence.The construction begins with a gravitational spacetime described by Einstein gravity.Source: B Spacetime Continuous Tensor Network · p. 29
  • A time slice foliated by codimension-two surfaces assigns a state to each surface, with the stated assumption that trivial homology gives a pure state.Source: B Spacetime Continuous Tensor Network · pp. 29–30
  • The proposed generator K(Σu) is expressed using local entangling operators Mi(x), with displacement of the surface specifying the evolution parameter.The operators are described as local up to the UV cutoff scale dual to the Planck length.Source: B Spacetime Continuous Tensor Network · pp. 29–30
  • Consistency across foliations yields a flat connection in the space of codimension-two surfaces, allowing K and the unitary matrix-valued function G(u) to define the network.The surface-dual state is written using G(u) acting on a reference state.Source: B Spacetime Continuous Tensor Network · p. 30
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