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

By Paweł Caputa, N. Kundu, Masamichi Miyaji, T. Takayanagi, Kento Watanabe. 2017

290 citations recorded by the source corpus.

A bstractWe 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e244c9c0d89f702e19ee2b0fd04d79413bd20fa0

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Citations in the corpus

Circuit complexity in quantum field theory (2017, background): “Indeed, it may be that the first steps in this direction have already been taken in [73, 74]—see also [75].”
Krylov complexity in conformal field theory (2021, background): “[38] Pawel Caputa, Nilay Kundu, Masamichi Miyaji, Tadashi Takayanagi, and Kento Watanabe, “Liouville action as path-integral complexity: from continuous tensor networks to ads/cft,” Journal of High Energy Physics 2017, 97 (2017).”
Holographic complexity in Vaidya spacetimes. Part II (2018, background): “uld be interesting to understand in which cases the quantum information metric could also be used to study the relative complexity of states before and after a global quantum quench. Another proposal [28,29,78] ties the complexity to the minimization of a functional given in terms of a generalized L”
Complexity and entanglement for thermofield double states (2018, methodology): “r important problem is to understand if there are relations between circuit complexity as considered here and complexities dened using the Fubini-Study metric [51] and the path-integral optimization [60,61,127]. Free QFTs provide what we believe is a fruitful testing ground in which all three approa”
Entanglement of purification through holographic duality (2017, background): “More generally, we can derive this inequality from equations (11) and (12).”