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Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, Kento Watanabe
TL;DR
The paper addresses how to optimize Euclidean path integrals that prepare CFT wave functionals. It optimizes the background metric, equivalently introducing a position-dependent UV cutoff, and finds hyperbolic geometry for two-dimensional CFT vacua. The prescription also reproduces entanglement wedges and holographic entanglement entropy when applied to reduced density matrices.
Problem
The paper asks how Euclidean path-integral representations of CFT states can be optimized while relating their geometry to tensor networks and holography.
Source: Appendix A · p. 6, Appendix B · p. 6Method
The authors vary the background metric or position-dependent cutoff and minimize the Liouville action, imposing Dirichlet and Neumann boundary conditions where appropriate.
Source: Appendix A · p. 6, Appendix B · p. 6Results
For two-dimensional CFT vacua, optimization yields hyperbolic geometry, while reduced-density-matrix optimization reproduces entanglement wedges and holographic entanglement entropy.
Source: Appendix A · p. 6, Appendix B · p. 6, Appendix C · p. 7Takeaways & Limitations
The prescription provides a geometric path-integral realization of tensor-network optimization and reproduces the standard entanglement-entropy result.
Source: Appendix A · p. 6, Appendix C · p. 7
Abstract
from arXiv · showhide
We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti--de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.
APPENDIX A: RELATION TO PATH INTEGRAL REPRESENTATIONS OF TENSOR NETWORKS
The appendix relates the optimization procedure to tensor-network path integrals by deforming the action through a position-dependent cutoff and minimizing the Liouville action. In a free scalar example, this yields the corresponding deformed on-shell action.
APPENDIX A: RELATION TO PATH INTEGRAL REPRESENTATIONS OF TENSOR NETWORKS
- The procedure deforms the Euclidean action through a position-dependent UV cutoff, generating a flow from the ground-state wave function toward an unentangled state.The deformation terminates the path integral at finite z values and is compared with continuous MERA.Source: Appendix A · p. 6
- Minimizing the Liouville action gives e2φ = 1.Source: Appendix A · p. 6
- The resulting coordinate-dependent-cutoff on-shell action agrees with the result obtained from the heuristic tensor-network argument [12].Source: Appendix A · p. 6
APPENDIX B: EXTRINSIC CURVATURES
The appendix defines extrinsic curvature for boundaries in the optimized metric and shows that vanishing curvature selects geodesic half circles in hyperbolic space. A boundary potential μB shifts this condition to nonzero curvature when replica geometry requires it.
APPENDIX B: EXTRINSIC CURVATURES
- The extrinsic curvature is defined by K = hab∇aNb, with components projected onto the boundary induced metric.Source: Appendix B · p. 6
- In hyperbolic space φ = −log z + const., circles z2 + x2 = l2 satisfy K = 0 for any l, identifying them as geodesics.Source: Appendix B · p. 6
- The boundary condition K + μB = 0 follows from varying the total Liouville action with respect to boundary shifts of φ.Source: Appendix B · p. 6
- Turning on µB gives the boundary condition with non-zero extrinsic curvature; along the cut at z = ǫ, eφ = 1/ǫ is imposed, while the half circle has Neumann condition (29).Source: Appendix B · p. 6
- Replica continuation changes the corner angle and shifts the curvature to K = π(n − 1), implemented by μB = π(1 − n).Source: Appendix B · p. 6
APPENDIX C: ANOTHER DERIVATION OF ENTANGLEMENT ENTROPY
The appendix provides an alternative replica derivation of entanglement entropy by combining curvature and boundary contributions in the optimized geometry. Evaluating the Liouville action reproduces the standard result.
APPENDIX C: ANOTHER DERIVATION OF ENTANGLEMENT ENTROPY
- Replica geometry produces conical-deficit contributions at the interval endpoints, including a scalar-curvature term proportional to 4π(1 − n).Source: Appendix C · p. 6
- The boundary extrinsic-curvature contribution is 2π(1 − n), keeping the total Euler number independent of n.Source: Appendix C · p. 6
- Terms proportional to n − 1 in the full Liouville action determine the entanglement entropy.Source: Appendix C · pp. 6–7
- The resulting entanglement entropy reproduces the well-known result.Source: Appendix C · p. 7