How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity Speaker(s): Bartek Czech
Abstract: According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the EinsteinHilbert action of a certain region in the dual spacetime. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries. By applying Tensor Network Renormalization to the discretized Euclidean path integral that prepares the CFT state, I will justify the recent suggestion by Caputa et al. that the complexity of a path integral is quantified by the Liouville action. The Liouville field specifies the conformal frame in which the path integral is evaluated; in the most efficient / least complexity frame, the Liouville field is closely related to entanglement entropies of CFT2 intervals. Assuming the RyuTakayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual spacetime. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesicwise rewriting of the nonlinear vacuum Einstein's equations in 3d with a negative cosmological constant. I emphasize that this is very much work in progress; I hope the audience will help me to sharpen the arguments.
Date: 20/04/2017  3:30 pm
Collection: Tensor Networks for Quantum Field Theories II
