Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT

1 Introduction

• Existing tensor-network arguments for AdS/CFT largely use discretized lattice models, motivating a continuous formulation applicable to genuine CFTs.cMERA extends the idea but has explicit formulations mainly for free theories, rather than strongly interacting holographic CFTs.Source: §1 · pp. 3–4
• The proposal provides a continuous limit of tensor networks while retaining field-theoretic computations and connects Euclidean path-integrals to TNR and MERA.TNR motivates viewing optimized Euclidean path-integrals as continuous tensor-network descriptions.Source: §1 · p. 4
• The approach keeps the Euclidean path-integral continuous while optimizing its lattice regularization through a position-dependent background metric.Scaling down the metric corresponds to coarse-graining under the rule that one lattice site occupies a unit area.Source: §1 · p. 4
• The functional I_Ψ measures path-integration size and is proposed as path-integral complexity, corresponding to computational complexity in the equivalent tensor network.For 2D CFTs, I_Ψ is identified with the Liouville action and minimized to define the state's complexity.Source: §1 · pp. 4–5
• The Euclidean construction avoids the ambiguity between de Sitter and hyperbolic interpretations and yields hyperbolic geometries identified with AdS time slices.This directly addresses a subtlety in interpreting MERA-like networks geometrically.Source: §1 · p. 5
• The paper studies 2D states, reduced density matrices, stress tensors, relative complexity, nearly conformal quantum mechanics, higher-dimensional CFTs, and time-dependent thermofield states.Its organization includes applications to entanglement wedges, higher-dimensional generalizations, and time evolution.Source: §1 · p. 5

2 Formulation of the Path-Integral Optimization

• Optimization means finding the most efficient discretized Euclidean path-integral that produces the correct quantum wave functional.The procedure is distinguished from optimizing tensor parameters; it changes the tensor-network structure through coarse-graining.Source: §2 · p. 6, §2.1 · p. 6

2.1 General Formulation

• Coarse-graining combines O(τ/ϵ) lattice sites after Euclidean evolution, producing the hyperbolic geometry shown in the optimized path-integral.Only low-energy modes survive over the evolution period, motivating the reduction in lattice sites.Source: §2.1 · p. 6, Figure 1 · p. 7
• A metric on path-integral space encodes the optimization by assigning one lattice site to a unit area while preserving the UV boundary conditions needed to recover the correct wave functional.In CFTs the background metric can be varied alone; nonconformal theories additionally require position-dependent external fields.Source: §2.1 · pp. 6–7
• The functional I_Ψ[g_ab] is minimized to define maximal optimization and is interpreted as the complexity of the partially optimized tensor network.The paper calls this quantity path-integral complexity and connects it to emergent AdS/CFT space.Source: §2.1 · p. 7, §2.2 · p. 9

2.3 Optimization of Vacuum States in 2D CFTs

• For a 2D vacuum, Weyl rescaling changes the path-integral normalization through the Liouville action, whose kinetic term represents the conformal anomaly and whose potential term reflects UV regularization.The wave functional remains proportional to the flat-metric result, with the proportionality coefficient determined by the Liouville action.Source: §2.3 · pp. 9–10
• Minimizing the Liouville action with the UV boundary condition yields the hyperbolic plane, identified with the AdS3 vacuum time slice for both free and interacting 2D CFTs.The optimized complexity scales linearly with the momentum cutoff ϵ^-1 and central charge c, matching the qualitative behavior of holographic complexity and the quantum information metric.Source: §2.3 · pp. 10–11

2.4 Tensor Network Renormalization and Optimization

• TNR motivates the identification because coarse-graining removes short-range entanglement, produces MERA, and reduces the number of tensors representing the Euclidean path-integral.In the continuous formulation, tensor-network insertions correspond to metric deformations, while the optimized path-integral corresponds approximately to MERA.Source: §2.4 · pp. 11–12
• The Liouville potential and kinetic terms respectively track tensor-volume and layer-gradient contributions, including unitary tensors, isometries, and disentanglers across TNR layers.The s-th MERA layer contributes the tensor counts associated with the layer geometry and is represented schematically in Figure 5.Source: §2.4 · pp. 11, 13, 14, Figure 5 · p. 13

3.1 Finite Temperature States

• Finite-temperature optimization produces a metric coinciding with the Einstein-Rosen bridge, the time slice of an eternal BTZ black hole.Source: §3.1 · pp. 14–15

3.2 CFT on a Cylinder and Primary States

• For vacuum states on a cylinder, the optimized geometry is the Poincare disk, which solves the Liouville equation.Source: §3 · p. 14, §3.2 · p. 15
• For primary states, the optimized conical geometry agrees with the gravity dual up to first-order corrections when hα ≪c.Source: §3.2 · p. 16
• Quantum Liouville theory gives perfect matching, while its large-c semiclassical approximation reproduces the classical optimization; the exact quantum proposal remains open.Source: §3.2 · p. 16

3.3 Liouville Equation and 3D AdS Gravity

• A nontrivial bulk coordinate transformation identifies the Liouville-theory appearance from optimization with the formulation arising in 3D AdS gravity.Source: §3.3 · pp. 16–17
• The optimized metric’s overall normalization, and therefore RAdS, is not fixed because it depends on the precise UV cutoff definition.Source: §3.3 · p. 17

4.1 Optimizing Reduced Density Matrices

• The optimization maps the plane with an interval cut to a smaller sphere with an open cut and minimizes the Liouville action subject to boundary conditions.Source: §4.1 · pp. 17–18, §4 · p. 17
• The optimized reduced-density-matrix geometry is two copies of the entanglement wedge pasted along the geodesic ΓA, which is the half circle z2+x2=l2.Source: §4.1 · pp. 17, 19, Figure 6 · p. 18
• Local operators inside the entanglement wedge survive optimization, whereas operators outside are washed out when back-reaction is neglected.Source: §4.1 · p. 19

4.2 Entanglement Entropy

• For n close to one, the replica construction introduces an infinitesimal negative deficit angle, implemented through a nonzero boundary Liouville potential.Source: §4.2 · p. 19, Figure 7 · p. 20
• The resulting entropy precisely agrees with the holographic entanglement entropy formula because ΓA is constrained to be geodesic.Source: §4.2 · pp. 19–20

4.3 Subregion Complexity

• The Liouville action evaluated on the optimized reduced subregion is proposed as a complexity measure for ρA, though more general states remain for future work.Source: §4.3 · p. 20

• Small-interval entanglement-entropy variations are related to Ttt through the first law, providing the basis for extracting the energy-momentum tensor.Source: §5 · p. 21
• For small state perturbations, the change in entropy is represented by a variation of the Liouville field and matched to the stress tensor.Source: §5 · p. 21
• The method reproduces the known stress tensors for thermofield-double and conical-singularity solutions, with a=1 yielding the Casimir energy.Source: §5 · pp. 21–22
• The resulting stress tensors agree with those obtained by varying the Liouville action with respect to the reference metric.Source: §5 · p. 22

6.1 General properties of the Liouville action

• The improved action obeys a chain rule and antisymmetry relation, while preserving the same equations of motion under variations of the Liouville field.Source: §6.1 · p. 24
• Because the Liouville action depends on the reference metric, the paper uses the relative functional IL[g2,g1] to measure complexity differences between path-integrals.Source: §6 · p. 22, §6.1 · pp. 24–25
• A constant Liouville-field shift changes the action when the Euler number is nonzero, requiring a background-charge prescription.Source: §6.1 · p. 25

6.2 Vacuum States

• For vacuum states, the evaluated actions for alternative metric representations differ, while the improved-action results satisfy the stated consistency relation.Source: §6.2 · pp. 25–26

6.3 Primary States

• For primary states, relative actions satisfy the same consistency relation, but conical geometries with a≠1 contain infrared divergences requiring an additional background charge.Source: §6.3 · pp. 26–27

6.4 Finite Temperature State

• For thermofield-double states, the relative actions evaluated in the Einstein-Rosen-bridge representations again satisfy the consistency relation.Source: §6.4 · pp. 27–28

• The conformal symmetry-breaking term is represented by a Schwarzian derivative action under reparameterizations, as in the SYK model.Source: §7 · p. 28
• For the one-dimensional metric ds2 = e2φdτ 2, the reparameterized time coordinate satisfies d˜τ/dτ = eφ, converting the breaking term into a φ-dependent action.Source: §7 · p. 28
• Minimizing the resulting action produces a geometry consistent with the time slice of AdS2 space-time.Source: §7 · p. 28
• The conformal symmetry-breaking term is necessary for stabilization, while the path-integral approach generates an extra dimension naturally even for one-dimensional quantum mechanics.Source: §7 · p. 28

8.1 Our Formulation

• The proposed higher-dimensional functional generalizes the Liouville action, with normalization fixed by holographic entanglement entropy and a d → 2 limit yielding relative Liouville complexity.Its optimized bulk-plus-boundary expression can be written covariantly in terms of the final metric, making reference-metric invariance manifest; higher-derivative terms are needed for correct even-dimensional anomalies.Source: §8.1 · pp. 29–30

8.2 AdS_d+1 from Optimization

• Optimizing the Weyl mode for vacuum CFT states produces hyperbolic space, coinciding with the time slice of pure AdS_d+1.The construction restricts optimization to Weyl scaling degrees of freedom, yet reproduces the expected geometry for CFTs on R^d and R×S^d−1.Source: §8 · p. 29, §8.2 · p. 31

8.3 Excitations in Global AdS_d+1

• Treating ˜ M as infinitesimally small, the optimization procedure correctly reproduces the first-order back-reaction to the AdS gravity time-slice metric, with h(r) matching η(r) up to a constant factor.Source: §8.3 · p. 32

8.4 Holographic Entanglement Entropy

• The optimization determines the bulk metric to be hyperbolic, with φ = −log z.Source: §8.4 · p. 34
• With normalization N chosen as in (8.3) and (8.4), the expressions reproduce the correct holographic entanglement-entropy behavior in higher dimensions, including the spherical-region formula, consistently with.Source: §8.4 · p. 35

8.5 Evaluation of Complexity Functional

• In any CFTd+1, the complexity scales as Ψ ∼ϵ−(d−1), a behavior interpreted as a volume-law divergence.Earlier proposals associated complexity with bulk volume through the CV-conjecture or with the action of a Wheeler-de Witt patch through the CA-conjecture [28] [29].Source: §8.5 · p. 36, §8.6 · p. 37

8.6 Comparison with Earlier Conjectures

• The proposal accesses only fixed-time complexity, so comparison with holographic prescriptions [26] [28] [29] [31] [34] is restricted to their constant-time evaluations.These comparisons concern divergence structures for the CV and CA conjectures, including the null-boundary prescription discussed in [31].Source: §8.6 · p. 38

Comparisons with Our Results

• CV has divergence structure c(1)v·ϵ−(d−3)+···, agreeing with CΨ0 but generally differing in relative coefficients.The comparison is made against results in (8.46), (8.44), and (8.45).Source: Comparisons with Our Results · p. 38
• CA gives log ϵ−1·ϵ(d−1), while [34] yields 1/ϵd−1 for d>2, matching the proposed divergence structure but generally differing numerically.For d=2, [34] instead makes the leading divergence vanish in Poincare and global AdS3, producing constant holographic complexity.Source: Comparisons with Our Results · p. 38

8.7 Relation to “Complexity = Action” Proposal

• Integrating the gravity action over the WDW geometry makes it proportional to Itot d[φ,g] up to AdS-boundary surface terms, with agreement holding on-shell.The identification uses a spacetime whose pure-AdS solution has a hyperbolic spatial metric; generic choices coincide with the WDW patch only on-shell.Source: §8.7 · p. 39
• In d=2, the proposal reduces to the Liouville action and retains a UV divergence O(ϵ−1), whereas the WDW action has no bulk contribution and does not.The resulting values are 0 for the CFT2 vacuum on R1 and π/4GN for the vacuum on S1, agreeing with [34].Source: §8.7 · p. 40

8.8 Higher Derivative Terms and Anomalies

• For four-dimensional CFTs, fourth-derivative terms are required to reproduce conformal anomalies and admit H4 as an AdS5 time slice, but the action is unbounded below.Thus the modified functional can be extremized rather than minimized, unlike the lower-derivative action (8.2).Source: §8.8 · pp. 40–41

9.1 Optimizing TFD State

• The TFD path integral optimizes its Euclidean segments by minimizing the Liouville action, while the Lorentzian evolution is assumed non-optimizable and placed between Euclidean segments.The rationale is that Euclidean evolution suppresses high-momentum sensitivity, whereas Lorentzian evolution retains sensitivity to the initial state.Source: §9.1 · p. 42, Figure 9 · p. 43
• The optimized TFD complexity grows linearly with time, consistent with half the WDW-patch gravity action and with holographic complexity expectations.The corresponding boundary-state example has half the TFD growth.Source: §9.1 · pp. 43–44

9.2 Comparison with Eternal BTZ black hole

• The Liouville-equation hyperbolic slice approaches a constant action, whereas the maximal-volume BTZ slice has nonconstant curvature and a volume term that grows linearly in time.The maximal-volume surface therefore matches the expected complexity growth more closely than the hyperbolic slice, but is not itself a Liouville solution.Source: §9.2 · p. 45
• For generic time-dependent backgrounds, identifying preferred gravity slices with Liouville solutions conflicts with using the Liouville action as complexity, leaving the extension unresolved.The paper explicitly treats the precise Liouville connection for time-dependent states as a future problem.Source: §9.2 · p. 46

9.3 Comment on Phase Transition

• The proposed phase-transition picture changes from a microscopic O(1) bridge at low temperature to a macroscopic O(c) bridge at high temperature, with a qualitative complexity crossing.The rough estimate gives C_dec < C_con for β ≪ 2π and the reverse for larger β, but predicts β = π^2/2 rather than the gravity value 2π.Source: Figure 10 · p. 46, §9.3 · pp. 46–47
• Comparing confinement and deconfinement complexities is not straightforward because only reference-metric-dependent differences are defined and the reference-space topology changes.The Liouville-theory Seiberg-bound interpretation nevertheless tracks the transition between confined and Einstein–Rosen-bridge geometries.Source: §9.3 · p. 47

10 Conclusions

• The paper defines Path-Integral Complexity by minimizing a metric-dependent functional for Euclidean CFT path integrals, identified with the Liouville action in two dimensions.Because of the conformal anomaly, two-dimensional complexity is treated through differences relative to a reference metric.Source: §10 · pp. 47–48
• Optimized metrics for static 2D CFT states coincide with time slices of expected gravity duals, supporting a continuous tensor-network interpretation of AdS3/CFT2.The framework also supplies a field-theoretic route to evaluating computational complexity and energy density.Source: §10 · p. 48
• Optimizing reduced density matrices produces two copies of the entanglement wedge, with boundary length reproducing holographic entanglement entropy.This connects the tensor-network geometry directly to the gravitational entanglement-wedge construction.Source: §10 · p. 48
• In one dimension, a conformal-symmetry-breaking effect yields a 1D Liouville action and explains an emergent extra dimension in NAdS2/CFT1.In higher dimensions, the proposed Weyl-mode functional reproduces vacuum time-slice metrics and spherical-subsystem holographic entanglement entropy.Source: §10 · pp. 48–49
• The higher-dimensional construction agrees with AdS-Schwarzschild geometry to first order in the mass parameter, but higher-order backreaction requires quantizing the Liouville mode.Higher-derivative corrections can preserve the stated properties, although the resulting action is no longer positive definite and must be extremized.Source: §10 · p. 49

A.1 Review of Earlier Argument

• The earlier 3D-gravity/Liouville construction maps Banados geometries to Liouville dynamics, while constant stress tensors produce BTZ black holes.The appendix establishes agreement between the Liouville stress tensor and the Banados-geometry stress tensor.Source: A Comparison with Earlier Liouville/3D… · p. 50, §A.1 · p. 51

A.2 Comparison with Our Approach

• The appendix expresses the Banados-to-Poincare-AdS3 map through holomorphic and antiholomorphic functions whose stress tensors determine the geometry.The displayed transformation provides the explicit coordinate relation between the two metric descriptions.Source: §A.2 · pp. 51–52
• The Euclidean Banados metric and the Liouville optimization ansatz are related by a coordinate transformation that is trivial near the AdS boundary.The corresponding Liouville and Banados energy-stress tensors agree.Source: §A.2 · pp. 51–52

B Holographic Complexity in the Literature

• The holographic-complexity review compares CV and CA prescriptions for Poincare and global AdS, including explicit WDW-action evaluations.The appendix fixes the AdS-radius convention to R_AdS = 1.Source: B Holographic Complexity in the… · p. 52
• For global AdS, the CA result contains a stronger leading logarithmic divergence associated with a joint term and the null-boundary parametrization ambiguity α.A prescription from [34] is cited as resolving that ambiguity.Source: B Holographic Complexity in the… · p. 53

C Higher Derivatives in Complexity Functional and Anomalies

• Assuming I4 = log Z4, the modified action reproduces the CFT4 trace anomaly structure and its equation of motion yields the hyperbolic time slice H4.The additional higher-derivative terms are related to the Weyl-squared and Euler-density contributions.Source: C Higher Derivatives in Complexity… · pp. 53–54
• The analysis restricts the optimization to Weyl-scaling metrics and extremizes the resulting action for Poincare and global AdS5 backgrounds.The action coefficients are parameterized through α3, α4, α5 and the metric determinant.Source: C Higher Derivatives in Complexity… · p. 54

C.1 Poincare AdS5 with Higher Derivatives

• For Poincare AdS5, e^φ = ℓ/z extremizes the higher-derivative action when the coefficients satisfy the stated condition involving a and b.The derivation assumes φ depends only on z and uses the action specialized to the Poincare time slice.Source: §C.1 · p. 54, C Higher Derivatives in Complexity… · p. 54

C.2 Global AdS5 with Higher Derivatives

• The global-AdS5 calculation uses the action specialized to a radial Weyl factor and checks the resulting equation of motion against the expected geometry.The metric is inserted into the higher-derivative action before extremization.Source: §C.2 · p. 55
• e^φ = 2ℓ/(1 − r2) solves the global AdS5 equation of motion, and the resulting condition matches the Poincare constraint, establishing the global time slice as an extremum.The calculation assumes φ depends only on r.Source: §C.2 · p. 55

C.3 Excitation in Global AdS5 with Higher Derivatives

• At first order in the black-hole mass M, the perturbed AdS5 Schwarzschild metric agrees with extremizing I4 when the zeroth-order coefficient condition is maintained.The solution is selected by imposing regularity at r = 1.Source: §C.3 · p. 55

D Entanglement Entropy and Liouville Field

• The regularized geodesic calculation can be reduced to twice the integral from the boundary to the geodesic turning point.This uses the symmetry of the geodesic and the regularized curve prescription.Source: D Entanglement Entropy and Liouville… · pp. 55–56
• For conformally transformed hyperbolic Liouville solutions, entanglement entropy is obtained from a line integral along the geodesic attached to the interval endpoints.The vacuum case gives the usual entropy when the geodesic length parameter equals the interval length.Source: D Entanglement Entropy and Liouville… · pp. 55–56
• The alternative positive-curvature Liouville description is related to the hyperbolic field by analytic continuation, but its physical significance remains unresolved.The paper nevertheless uses the first law to compute the stress tensor in the small-interval expansion.Source: D Entanglement Entropy and Liouville… · pp. 55–56
• The stress tensor can be extracted from the transformed Liouville field through an equation involving the Schwarzian derivative of the conformal map.The same construction applies analogously to the antiholomorphic function.Source: D Entanglement Entropy and Liouville… · p. 57