Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT

TL;DR

The paper develops a path-integral optimization framework for conformal field theories, identifying a state- and dimension-dependent complexity functional that, in 2D, reduces to the Liouville action. By minimizing this functional over background metrics, the authors derive hyperbolic time slices matching AdS/CFT expectations and recover holographic entanglement entropy from optimized reduced density matrices. The approach unifies continuous tensor networks with AdS/CFT, extends to higher dimensions with a generalized Weyl-mode functional, and connects to holographic complexity concepts, including comparisons to Complexity=Volume and Complexity=Action. It also explores time dependence via TFD evolution, phase transitions, and higher-derivative corrections to capture conformal anomalies, offering a concrete field-theoretic route to emergent bulk geometry and complexity measures.

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.

Figures (10)

• Figure 1: Computation of a ground state wave function from Euclidean path-integral (left) and its optimization (middle), which is described by a hyperbolic geometry. The right figure schematically shows its tensor network expression.
• Figure 2: A quantum circuit representation for a quantum state in a qubit system. A quantum state $| \Psi \rangle$ can be constructed by simple local (2-qubit) unitary operations from a simple reference state, for example, a product state $|0\rangle |0\rangle |0\rangle \cdots$.
• Figure 3: The Euclidean path integral for the ground state wave functional $\Psi[\tilde{\varphi}(x)]$ can be approximately described by a tensor network on a square lattice.
• Figure 4: The tensor network renormalization (TNR) gradually makes the coarse-grained tensor network with removing short-range entanglement. From the UV boundary, isometries (coarse-graining) and unitaries (disentanglers) accumulate and the MERA network grows with the TNR steps.
• Figure 5: The tensor network produced when we have a shift of $\phi$ at a specific layer. This also represents the one step ($s$-th) contribution in the process of tensor network renormalization, which finally reaches the MERA network. This corresponds to $s-$th terms $\int^{2^s \epsilon}_{2^{s-1} \epsilon} dz(\cdot\cdot\cdot)$ in (\ref{['tnrcomp']}).