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Journal of High Energy Physics

, 2017:97 | Cite as

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

  • Pawel Caputa
  • Nilay Kundu
  • Masamichi Miyaji
  • Tadashi Takayanagi
  • Kento Watanabe
Open Access
Regular Article - Theoretical Physics

Abstract

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 arXiv: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.

Keywords

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

Notes

Open Access

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.

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Pawel Caputa
    • 1
  • Nilay Kundu
    • 1
  • Masamichi Miyaji
    • 1
  • Tadashi Takayanagi
    • 1
    • 2
  • Kento Watanabe
    • 1
  1. 1.Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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