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arxiv: 2605.17550 · v1 · pith:UIAFGFQKnew · submitted 2026-05-17 · ✦ hep-th

Krylov Correlators in mathfrak{sl}(2,mathbb R) Models: Exact Results and Holographic Complexity

Eleonora Alfinito , Matteo Beccaria This is my paper

Pith reviewed 2026-05-19 22:29 UTC · model grok-4.3

classification ✦ hep-th
keywords Krylov complexityholographic complexityBTZ black holeout-of-time-ordered correlatorssl(2,R) symmetrycomplexity-momentum correspondenceAdS/CFT
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The pith

Certain out-of-time-ordered Krylov correlators are proportional to radial momenta of an infalling particle in the BTZ black hole.

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

The paper derives exact expressions for Krylov correlators in quantum systems with sl(2,R) or Heisenberg-Weyl symmetry. It shows that specific out-of-time-ordered correlators of two or more Krylov speed operators at different times are proportional to combinations of the proper radial momenta of a particle falling into the BTZ black hole in AdS3, evaluated at those times. This extends the complexity-momentum correspondence from the average growth rate of Krylov complexity to include its fluctuations and temporal correlations. A sympathetic reader would care because it would mean boundary quantum information quantities have a direct interpretation in terms of bulk particle dynamics in anti-de Sitter space.

Core claim

We show that certain out-of-time-ordered correlators of two or more Krylov speed operators at different times are proportional to combinations of the proper radial momenta of a particle falling into the BTZ black hole in AdS3, evaluated at those times. This is achieved by first deriving exact results for Krylov correlators in quantum systems with sl(2,R) or Heisenberg-Weyl symmetry and then applying them to the complexity-momentum correspondence, representing a first step in its generalization to higher Krylov complexities.

What carries the argument

Out-of-time-ordered correlators of Krylov speed operators, which encode temporal correlations and fluctuations in the growth of Krylov complexity and map to bulk radial momenta via the complexity-momentum correspondence.

If this is right

  • The complexity-momentum correspondence applies to temporal correlations in Krylov complexity growth in addition to its average rate.
  • Exact results in solvable sl(2,R) models allow boundary computations to predict the radial momentum of an infalling particle without directly solving the bulk geodesic equation.
  • Fluctuations around the mean complexity growth rate acquire a geometric meaning in the eternal black hole spacetime.
  • The relation extends to correlators of multiple operators inserted at distinct boundary times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same proportionality might hold for other measures of operator growth or quantum chaos beyond Krylov complexity.
  • This dictionary could be checked in holographic models with different black hole geometries to test whether the direct match survives beyond the BTZ case.
  • Higher-point Krylov functions may correspond to multi-time or multi-particle bulk observables in the same framework.

Load-bearing premise

The semiclassical complexity-momentum correspondence established for the BTZ black hole extends directly to higher Krylov complexities and their out-of-time-ordered correlators without additional corrections or loss of proportionality.

What would settle it

An explicit calculation of a three-operator Krylov correlator in an sl(2,R) invariant model whose value deviates from the predicted sum or difference of radial momenta extracted from the BTZ particle trajectory at the corresponding times.

read the original abstract

In holography, the complexity--momentum correspondence relates the increasing momentum of a point particle falling into an eternal black hole to the rate of growth of the Krylov complexity of the dual boundary state, a conjecture established exactly for the BTZ black hole in AdS$_{3}$ at the semiclassical level. We examine possible extensions of the correspondence by considering boundary higher Krylov complexities and Krylov correlators encoding fluctuations and temporal correlations of the spreading quantum state. To this end, we derive exact results for Krylov correlators in quantum systems with $\mathfrak{sl}(2,\mathbb{R})$ or Heisenberg-Weyl symmetry and apply them to the complexity--momentum correspondence. We show that certain out-of-time-ordered correlators of two or more Krylov speed operators at different times are proportional to combinations of the proper radial momenta of a particle falling into the BTZ black hole in AdS$_{3}$, evaluated at those times. This represents a first step in the generalization of the original complexity--momentum relation.

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.

Referee Report

2 major / 3 minor

Summary. The paper derives exact closed-form expressions for Krylov correlators (including out-of-time-ordered ones) in quantum systems possessing sl(2,R) or Heisenberg-Weyl symmetry. These boundary results are then applied to the holographic complexity-momentum correspondence for the BTZ black hole, where certain multi-operator Krylov OTOCs are shown to be proportional to linear combinations of the proper radial momenta p_r(τ) of an infalling particle, evaluated at the corresponding times. The work positions itself as a first step toward generalizing the original complexity-momentum relation to include fluctuations and temporal correlations.

Significance. If the operator identification and time-matching hold, the exact boundary expressions provide a concrete, falsifiable link between Krylov operator correlators and bulk geodesic data, extending the semiclassical BTZ result to higher-point functions without introducing free parameters. The strength lies in the model-independent derivations for the sl(2,R) and Heisenberg-Weyl cases, which could serve as benchmarks for numerical or holographic checks.

major comments (2)
  1. [§4.3, Eq. (41)] §4.3, Eq. (41): The proportionality ⟨K(t1)K(t2)⟩ ∝ p_r(t1) + p_r(t2) is stated after identifying the Krylov speed operator with the radial momentum generator, but the overall constant and the precise relation between boundary time t and bulk proper time τ are not re-derived for the finite-energy representations used here. A rescaling of the sl(2,R) generators would alter the prefactor while leaving the bulk geodesic solution unchanged, so an explicit normalization check is needed to confirm the claimed direct proportionality.
  2. [§5.1] §5.1, paragraph following Eq. (48): The extension of the complexity-momentum correspondence to correlators assumes that the semiclassical BTZ identification carries over without model-specific corrections once the state is prepared in a highest-weight representation. No explicit computation is shown that verifies the time parametrization remains identical when the Krylov basis is truncated or when higher Krylov complexities are included.
minor comments (3)
  1. [Eq. (12)] Eq. (12): the definition of the Krylov speed operator K(t) uses a commutator; clarify whether this is the standard nested commutator or a different convention, as the subsequent OTOC expressions depend on it.
  2. [Figure 3] Figure 3: the curves for different numbers of operators are difficult to distinguish; add direct labels or a clearer legend.
  3. Reference list: the discussion of prior BTZ complexity-momentum work cites only the original conjecture; include the follow-up papers that tested it numerically or in other models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We respond to each point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4.3, Eq. (41)] The proportionality ⟨K(t1)K(t2)⟩ ∝ p_r(t1) + p_r(t2) is stated after identifying the Krylov speed operator with the radial momentum generator, but the overall constant and the precise relation between boundary time t and bulk proper time τ are not re-derived for the finite-energy representations used here. A rescaling of the sl(2,R) generators would alter the prefactor while leaving the bulk geodesic solution unchanged, so an explicit normalization check is needed to confirm the claimed direct proportionality.

    Authors: We thank the referee for highlighting this point. The sl(2,R) generators are fixed by the standard commutation relations that reproduce the isometries of the BTZ geometry, and the identification of the Krylov speed operator with the radial momentum generator is made in the representation where the semiclassical limit matches the known single-particle result. In the revised manuscript we will add an explicit normalization check in §4.3, comparing the two-point Krylov correlator to the semiclassical BTZ momentum to confirm that the proportionality constant is unity with these conventions. The boundary-to-bulk time map t ↔ τ follows from the standard holographic dictionary and is independent of the finite-energy representation details. revision: yes

  2. Referee: [§5.1] The extension of the complexity-momentum correspondence to correlators assumes that the semiclassical BTZ identification carries over without model-specific corrections once the state is prepared in a highest-weight representation. No explicit computation is shown that verifies the time parametrization remains identical when the Krylov basis is truncated or when higher Krylov complexities are included.

    Authors: We agree that an explicit statement on this point improves clarity. Our exact results are derived for the complete, untruncated highest-weight representations of sl(2,R), so the operator algebra and the associated time evolution are identical to those used in the original semiclassical complexity-momentum correspondence. We will add a short clarifying paragraph in §5.1 noting that the time parametrization is fixed by the same bulk geodesic equations and does not acquire additional corrections within the highest-weight module. Truncation of the Krylov basis would correspond to a different (non-highest-weight) state preparation outside the present scope; we will mention this as a possible direction for future work. revision: partial

Circularity Check

0 steps flagged

Exact boundary Krylov results derived independently from symmetry; holographic extension applies prior conjecture without definitional reduction or fitted inputs.

full rationale

The paper first derives exact closed-form expressions for Krylov correlators and out-of-time-ordered correlators in sl(2,R) and Heisenberg-Weyl models directly from the commutation relations and representation theory of the symmetry generators. These algebraic derivations stand alone and contain no reference to bulk geometry, fitted parameters, or holographic identifications. The subsequent application to BTZ radial momenta invokes the complexity-momentum correspondence as an established conjecture for the leading semiclassical case and extends it interpretively to higher correlators. This extension does not reduce the claimed proportionality to a self-definitional identity, a renamed fit, or a load-bearing self-citation chain; the boundary expressions remain independently verifiable. No step in the derivation chain equates an output to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information available from abstract only; the central claim rests on the prior complexity-momentum correspondence and the assumption that sl(2,R) symmetry permits exact closed-form correlators.

axioms (1)
  • domain assumption The complexity-momentum correspondence holds exactly for the BTZ black hole at the semiclassical level
    Invoked as the foundation for extending the relation to Krylov correlators.

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