The Holography of Spread Complexity: A Story of Observers

Abstract

Building on the pioneering work of \cite{Caputa:2024sux}, we propose a holographic description of spread complexity and its rate in 2D CFTs. By exploiting $SL(2,\mathbb{R})$ symmetry, we explicitly construct the Krylov basis, expressing spread complexity as a linear combination of generator expectation values. Within the AdS/CFT correspondence, we translate these boundary expectations directly into bulk kinematic variables. These findings suggest that spread complexity manifests as the energy measured by a bulk observer, with its rate corresponding to the radial momentum.

version 3. Substantially rewrite the Abstract, Introduction, and Section 2, with minor revisions to the remaining sections. The misunderstanding regarding the ambiguity arising from the stabilizer group has been removed. Expand the discussion on the setup, emphasizing how the extrapolated AdS/CFT dictionary is utilized to derive the holographic expressions for spread complexity. Refs are added