The Holography of Spread Complexity: A Story of Observers

Zhehan Li , Jia Tian

Authors on Pith no claims yet

Pith reviewed 2026-05-05 05:02 UTC · model claude-opus-4-7

classification ✦ hep-th

keywords complexityspreadbulkratebasisboundarybuildingcaputa

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The pith

Spread complexity in 2D CFTs is the energy a bulk observer measures, and its growth rate is that observer's radial momentum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks what spread complexity — a tractable measure of how a quantum state spreads through Hilbert space under time evolution — actually corresponds to in the dual gravity picture. Using SL(2,R) symmetry of a 2D CFT, the authors construct the Krylov basis explicitly without running the Lanczos algorithm and show that the complexity of a state created by a primary operator is a specific linear combination of SL(2,R) generator expectation values. They then push this combination through the extrapolated AdS/CFT dictionary: each generator becomes a Killing vector in the bulk, and each expectation value becomes the projection of the dual particle's momentum onto that Killing vector. The complexity itself comes out as a tetrad projection — the energy of the particle as seen by a definite observer — while its rate comes out as the radial component of the same observer's measured momentum. This sharpens an earlier "spread complexity rate equals proper momentum" conjecture by replacing a coordinate-dependent radial momentum with a tetrad-projected, coordinate-invariant one. They check the picture in global, Poincaré, and Rindler AdS3, and argue the construction lifts to higher dimensions whenever an effective SL(2,R) controls the dynamics.

Core claim

For a primary scalar state in a 2D CFT with SL(2,R) symmetry, the authors build the Krylov basis directly from the representation theory rather than iteratively, and show that spread complexity is a fixed linear combination of expectation values of the SL(2,R) generators. Translating those boundary expectation values into the bulk via the extrapolated AdS/CFT dictionary, the generator expectations become projections of the dual particle's four-momentum onto a tetrad built from the Killing fields. The conclusion they argue for: spread complexity is the energy of the bulk particle as measured by a particular observer, and its time derivative is the radial momentum that observer measures. Becau

What carries the argument

The SL(2,R) coherent state representation of primary states O(z)|Ω⟩, identifying them as highest-weight states of a rotated sl2 algebra {L0, L±1}. This lets the Krylov basis be written as Ln_{-1}|∆⟩_z and the complexity as ⟨L0⟩ − ∆, which under the extrapolated dictionary becomes a tetrad-projected bulk momentum ê·p along Killing vectors associated with the generators.

If this is right

<parameter name="0">The coordinate ambiguity in the "complexity rate equals radial momentum" conjecture is removed: the right object is the tetrad-projected momentum
not any particular coordinate component.

Load-bearing premise

That the bulk-boundary identification valid only as the operator approaches the AdS boundary (the extrapolated dictionary) can be extended to operators inserted anywhere inside the disk, treating the dual as a single classical particle on a geodesic for a generic insertion point.

What would settle it

Compute spread complexity directly from Lanczos coefficients for a primary state inserted at a finite, non-boundary point z0 in a holographic 2D CFT (e.g. heavy operator with 1 ≪ ∆ ≪ 1/G_N), and compare to ê0·p evaluated on the proposed bulk geodesic. If the two disagree, or if the rate fails to match the tetrad-projected radial momentum in any of the three coordinate systems (global, Poincaré, Rindler) considered, the proposed identification fails.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

pith-pipeline@v0.1.0 · 4483 in / 1421 out tokens · 22200 ms · 2026-05-05T05:02:20.316692+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Holographic Krylov Complexity for Charged, Composite and Extended Probes
hep-th 2026-04 unverdicted novelty 7.0

Holographic Krylov complexity for charged composite and extended probes retains universal leading large-time growth but acquires structure-dependent subleading corrections.
Probing the Chaos to Integrability Transition in Double-Scaled SYK
hep-th 2026-01 unverdicted novelty 5.0

A first-order phase transition in the Berkooz-Brukner-Jia-Mamroud interpolating model causes chord number, Krylov complexity, and operator size to switch discontinuously from chaotic (linear/exponential) to quasi-inte...