JHEP Journal of High Energy Physics 1029-8479 JHEP11(2017)097 10.1007/JHEP11(2017)097 Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT Pawel Caputa pawel.caputa@yukawa.kyoto-u.ac.jp Nilay Kundu nilay.kundu@yukawa.kyoto-u.ac.jp Masamichi Miyaji masamichi.miyaji@yukawa.kyoto-u.ac.jp Tadashi Takayanagi takayana@yukawa.kyoto-u.ac.jp Kento Watanabe wkento@yukawa.kyoto-u.ac.jp Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP),Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo,Kashiwano-ha, Kashiwa, Chiba 277-8582, Japan 16112017 2017 11 097 28082017 26102017 OPEN ACCESS, © The Authors 2017 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 1706.07056

We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report https://arxiv.org/abs/1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

AdS-CFT Correspondence Anomalies in Field and String Theories Conformal Field Theory Holography and condensed matter physics (AdS/CMT)

Article funded by SCOAP3

 Introduction

Formulation of the path-integral optimization

General formulation

Connection to computational complexity

Optimization of vacuum states in 2D CFTs

Tensor network renormalization and optimization

Optimizing various states in 2D CFTs

Finite temperature states

CFT on a cylinder and primary states

Liouville equation and 3D AdS gravity

Reduced density matrices and EE

Optimizing reduced density matrices

Entanglement entropy

Subregion complexity

Energy momentum tensor in 2D CFTs

Evaluation of \texorpdfstring{\boldmath $S_L$}{S(L)} in 2D CFTs

General properties of the Liouville action

Vacuum states

Primary states

Finite temperature state

Application to NAdS\texorpdfstring{\boldmath $_2/$}{(2)/}CFT\texorpdfstring{\boldmath $_1$}{(1)}

Applications to higher dimensional CFTs

Our formulation

\boldmath AdS$_{d+1}$ from optimization

\boldmath Excitations in global AdS$_{d+1}$

Holographic entanglement entropy

Evaluation of complexity functional

Comparison with earlier conjectures

2$ is the same as ours, whereas, they differ in the numerical coefficients in general. Nevertheless, in the next subsection, we will point out an interesting relation between our complexity functional $I^{\rm tot}_d$ and the gravity action $I_{\rm WDW}$ in the WDW patch. Since there is no precise definition of computational complexity in quantum field theories known at present, we cannot decide which of these prescriptions is true by consulting with rigorous results in field theory. However, notice that our proposal of computational complexity $C_{\Psi_0}$, defined in~(\ref{cpxidef}), is based on not any holography but a purely field theoretic argument as is clear in two dimensional CFT case, where it is related to the normalization of wave functional. ]]>

 Relation to ``complexity = action'' proposal

Higher derivative terms and anomalies

Discussions: time evolution of TFD states and phase transitions

Optimizing TFD state

Comparison with eternal BTZ black hole

Comment on phase transition

Conclusions

Acknowledgments

Comparison with earlier Liouville/3D gravity relation

Review of earlier argument

Comparison with our approach

Holographic complexity in the literature

Higher derivatives in complexity functional and anomalies

\boldmath Poincar\'e AdS$_5$ with higher derivatives

\boldmath Global AdS$_5$ with higher derivatives

\boldmath Excitation in global AdS$_5$ with higher derivatives

Entanglement entropy and Liouville field

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