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.
In 2D CFTs with SL(2,R) symmetry, spread complexity can be written as a linear combination of SL(2,R) generator expectation values, which under AdS/CFT translate to the energy measured by a bulk observer, while its rate corresponds to the radial momentum.
That the SL(2,R)-symmetric Krylov construction (extending Caputa et al.) generalizes to a genuine holographic dictionary, and that the identification of generator expectation values with bulk kinematic observables (observer energy, radial momentum) is unique rather than one of several gauge/frame-dependent choices.
1/ Spread complexity measures how an evolving quantum state spreads over a special (Krylov) basis. The authors build this basis explicitly in 2D CFTs using SL(2,R) symmetry. 2/ They rewrite spread complexity as a sum of expectation values of SL(2,R) generators, then use AdS/CFT to translate these boundary quantities into bulk geometric variables. 3/ The punchline: complexity looks like the energy a bulk observer measures, and its time derivative looks like radial momentum — a concrete entry in the complexity=geometry dictionary, restricted to a symmetric setting.
They show a math measure of how a quantum state spreads acts like the energy seen by someone inside curved space.
Abstract-only review. The proposal extends Caputa:2024sux by exploiting SL(2,R) to make the Krylov basis explicit and then mapping generator expectation values to bulk kinematics. This is a plausible and active line in holographic complexity, but I cannot verify the construction of the Krylov basis, the linear-combination formula, or the bulk identification without the full text. The identification of complexity with observer energy echoes recent observer-based holography work (de Sitter observers, CLPW), so novelty is incremental rather than foundational. Confidence is LOW given abstract-only access.