Lorentzian Threads and Holographic Complexity

 The continuous min flow-max cut principle is used to reformulate the 'complexity=volume' conjecture using Lorentzian flows. Conceptually, discretized flows are interpreted in terms of `gatelines', one-dimensional time-like curves that connect layers of a tensor network grid in the bulk spacetime. We imagine each gateline represents a unitary operation such that the bulk calculation for complexity matches its information-theoretic definition. The bulk symplectic potential provides a 'canonical' flow configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of 'spacetime complexity'. Finally, we explain the need for a more general measure of complexity that captures the role of suboptimal flows or tensor network configurations. Based on 2105.12735 and 2106.12585.