The Holography of Spread Complexity: A Story of Observers

TL;DR

This paper develops a holographic framework for spread complexity in 2D CFTs, linking boundary expectations to bulk kinematic variables and interpreting complexity as energy measured by a bulk observer.

Contribution

It introduces a holographic description of spread complexity using $SL(2, ext{R})$ symmetry and constructs the Krylov basis explicitly within this framework.

Findings

Spread complexity is expressed as a linear combination of generator expectation values.

Boundary expectations are translated into bulk kinematic variables.

Spread complexity corresponds to the energy measured by a bulk observer.

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.