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

Pawel Caputa,Tadashi Takayanagi,Nilay Kundu,Kento Watanabe,Masamichi Miyaji

doi:10.1007/jhep11(2017)097

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.

Highlights

The AdS/conformal field theories (CFTs) correspondence [1] has been the most powerful tool to understand quantum nature of gravity
We will find that it depends on the reference metric and it does not seem to be possible to define its absolute value, which is due to the conformal anomaly in 2D CFTs
As we argued for 2D CFTs (see discussions following (2.13)), the Liouville action, SL when computed on-shell for the solutions gives us a measure of holographic computational complexity

Summary

Introduction

The AdS/CFT correspondence [1] has been the most powerful tool to understand quantum nature of gravity. Up to now, most arguments in these directions have been limited to studies of discretized lattice models so that we can apply the idea of tensor networks directly. They at most serve as toy models of AdS/CFT as they do not describe the genuine CFTs which are dual to the AdS gravity (though they provide us with deep insights of holographic principle such as quantum error corrections [10, 17, 18]). There already exists a formulation called cMERA (continuous MERA) [5], whose connection to

Objectives

Results

Conclusion