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arxiv: 2506.01957 · v3 · submitted 2025-06-02 · ✦ hep-th · cond-mat.str-el· quant-ph

Violation of Universal Operator Growth Hypothesis in mathcal{W}₃Conformal Field Theories

Dileep P. Jatkar , Sujoy Mahato , Sukrut Mondkar , Praveen Thalore This is my paper

Pith reviewed 2026-05-19 11:09 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords W3 conformal field theoryoperator growthLanczos coefficientsKrylov complexityhigher-spin symmetryinformation scramblinguniversal operator growth hypothesis
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The pith

In W3 conformal field theories a generalized Liouvillian that includes higher-spin generators produces quadratic growth in Lanczos coefficients and violates the universal operator growth hypothesis.

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

The paper shows that operator growth in large-central-charge conformal field theories with W3 symmetry exceeds the linear growth bound once the Liouvillian is enlarged to probe spin-3 generators. For the combination of Virasoro and W modes acting on the descendant module of a heavy primary, several sectors of the Lanczos coefficients grow quadratically with level N. This superlinear growth makes Krylov complexity diverge and demonstrates that extended higher-rank symmetries can change scrambling dynamics beyond what standard bounds assume. A reader would care because the result challenges the expectation that operator growth follows universal rules across all conformal theories.

Core claim

In large-central-charge W3 conformal field theories the Lanczos coefficients for the generalized Liouvillian L = κ1 (L1 + L-1) + κ2 (W2 + W-2) include sectors with asymptotic behavior b_N ~ N^2 when computed in the descendant module of a heavy primary. This quadratic growth exceeds the conjectured bound of the universal operator growth hypothesis and renders Krylov complexity divergent. The same quadratic growth already appears when the analysis is restricted to the global SL(3,R) subalgebra, showing that the violation is rooted in the extended symmetry itself.

What carries the argument

The generalized Liouvillian that mixes Virasoro generators with the spin-3 W generators, whose repeated action on the heavy-primary descendant module produces the Lanczos coefficients that control the growth rate.

If this is right

  • The universal operator growth hypothesis fails to hold once higher-spin generators are included in the Liouvillian.
  • Krylov complexity diverges in these W3 theories because of the superlinear Lanczos growth.
  • The violation is already present in the finite-dimensional global SL(3,R) subalgebra and is therefore intrinsic to the higher-rank symmetry.
  • Extended W-symmetries can qualitatively alter operator growth and evade the conventional bounds that apply to Virasoro theories.

Where Pith is reading between the lines

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

  • Holographic models with higher-spin gravity may therefore exhibit faster scrambling than their Virasoro counterparts.
  • Universal bounds on operator growth will need to be revised or conditioned on the rank of the symmetry algebra.
  • Analogous quadratic growth could appear in other W-algebras or in CFTs possessing additional conserved higher-spin currents.

Load-bearing premise

The Lanczos coefficients obtained from the generalized Liouvillian acting on the descendant module of one heavy primary continue to represent the operator growth of the full theory without being changed by the remaining infinite tower of higher-spin generators.

What would settle it

An explicit calculation of the Krylov basis in a specific large-central-charge W3 model that shows only linear growth in the Lanczos coefficients would falsify the quadratic-growth claim.

read the original abstract

We show that operator growth in large-central-charge conformal field theories with $\mathcal{W}_3$ symmetry can violate the universal operator growth hypothesis once the Liouvillian is enlarged to probe the higher-spin generators. For the generalized Liouvillian $\mathcal{L} = \kappa_1 \left( L_1 + L_{-1} \right) + \kappa_2 \left( W_2 + W_{-2} \right)$, we compute the Lanczos coefficients in the descendant module of a heavy primary and find several classes with faster-than-linear growth in the descendant level $N$, including maximally violating sectors with asymptotic behavior $b_N \sim N^2$. This superlinear growth exceeds the conjectured bound and renders the Krylov complexity divergent. We further show that the same quadratic asymptotic growth already arises in the global $SL(3, \mathbb{R})$ subalgebra, indicating that the violation is rooted in the extended higher-rank symmetry itself. Our results demonstrate that extended $\mathcal{W}$-symmetries can qualitatively modify operator growth and evade conventional bounds on information scrambling.

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 / 2 minor

Summary. The paper claims that operator growth in large-central-charge W3 CFTs violates the universal operator growth hypothesis when the Liouvillian is extended to include higher-spin generators. For L = κ1(L1 + L-1) + κ2(W2 + W-2), the Lanczos coefficients b_N extracted from the action on the descendant module of a heavy primary include sectors with asymptotic quadratic growth b_N ~ N^2, exceeding the conjectured linear bound and rendering Krylov complexity divergent. The same quadratic growth is reported already within the global SL(3,R) subalgebra, suggesting the violation originates from the extended higher-rank symmetry.

Significance. If the asymptotic analysis holds, the result shows that W-extended symmetries can qualitatively modify operator growth and evade standard bounds on scrambling, with implications for chaos and complexity in higher-spin AdS3/CFT2. A strength is the explicit verification of quadratic growth inside the finite-dimensional global SL(3,R) subalgebra, which provides a controlled, algebraically closed setting for the claim.

major comments (2)
  1. [Lanczos coefficient computation] The central claim of b_N ~ N^2 in certain sectors rests on the three-term Lanczos recurrence obtained from matrix elements of the generalized Liouvillian between normalized descendants. The W3 algebra includes non-vanishing [W_m, W_n] commutators that produce both Virasoro and lower-spin modes plus central-charge terms at every level; the manuscript must demonstrate explicitly that these relations are retained without truncation or large-c approximations that could cancel the leading quadratic term (see the recursion in the descendant module section).
  2. [Global SL(3,R) subalgebra analysis] Even in the global SL(3,R) subalgebra, the induced module on a primary is infinite-dimensional and the orthogonalization step uses the full sl(3) commutation relations. Any reordering or central-term contributions at high descendant levels could reduce the effective growth below quadratic; the paper should supply the explicit recurrence coefficients or a closed-form argument showing that no such cancellation occurs.
minor comments (2)
  1. [Introduction] The parameters κ1 and κ2 are introduced by hand to define the generalized Liouvillian; a brief discussion of their physical interpretation or limiting cases would improve clarity.
  2. [Numerical results] Ensure that any plots of b_N versus N include the fitting procedure, the range of N used for the asymptotic extraction, and error estimates from the orthogonalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the presentation of our results on operator growth in W3 CFTs. We address each major comment point by point below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Lanczos coefficient computation] The central claim of b_N ~ N^2 in certain sectors rests on the three-term Lanczos recurrence obtained from matrix elements of the generalized Liouvillian between normalized descendants. The W3 algebra includes non-vanishing [W_m, W_n] commutators that produce both Virasoro and lower-spin modes plus central-charge terms at every level; the manuscript must demonstrate explicitly that these relations are retained without truncation or large-c approximations that could cancel the leading quadratic term (see the recursion in the descendant module section).

    Authors: We agree that an explicit verification of the full algebra is essential. Our computations in the manuscript employ the complete W3 commutation relations, including all [W_m, W_n] terms that generate Virasoro modes, lower-spin contributions, and central-charge corrections, without truncation or large-c cancellations that would eliminate the quadratic growth. The leading N^2 behavior originates from the action of the higher-spin generators on the normalized descendants and persists asymptotically. To strengthen the presentation, we will add an expanded derivation in the revised manuscript (as a new appendix) that walks through the matrix elements for representative sectors, explicitly retaining every commutator contribution and showing why subleading terms do not cancel the quadratic asymptotics. revision: yes

  2. Referee: [Global SL(3,R) subalgebra analysis] Even in the global SL(3,R) subalgebra, the induced module on a primary is infinite-dimensional and the orthogonalization step uses the full sl(3) commutation relations. Any reordering or central-term contributions at high descendant levels could reduce the effective growth below quadratic; the paper should supply the explicit recurrence coefficients or a closed-form argument showing that no such cancellation occurs.

    Authors: We appreciate this observation on the global subalgebra. Within SL(3,R), the module is infinite-dimensional and we use the exact commutation relations for orthogonalization. Our explicit computations already demonstrate that reordering and central terms remain subdominant, preserving the quadratic growth b_N ~ N^2. To address the request directly, the revised manuscript will include the explicit recurrence coefficients for the first several levels together with an inductive argument establishing that no cancellation alters the leading quadratic coefficient at high N. This closed-form reasoning confirms the growth is intrinsic to the higher-rank algebra structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Lanczos coefficients derived from explicit algebra action on module

full rationale

The derivation proceeds by defining an explicit generalized Liouvillian with free parameters κ1, κ2 and then computing the three-term Lanczos recurrence directly from matrix elements of this operator between normalized descendants in the W3 module (and separately in the finite-dimensional global SL(3,R) subalgebra). The reported b_N ∼ N^2 asymptotics are outputs of that recursion, not inputs or fitted quantities; the algebra commutation relations are used as external structure rather than being redefined in terms of the growth rates. No self-citation is load-bearing for the central claim, no ansatz is smuggled, and the result is not a renaming of a known pattern. The computation is therefore self-contained against the stated algebraic inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard CFT axioms, the definition of the generalized Liouvillian with two free coefficients, and the conjecture being violated; no new entities are postulated.

free parameters (2)
  • κ1
    Coefficient weighting the Virasoro generators in the generalized Liouvillian.
  • κ2
    Coefficient weighting the W generators in the generalized Liouvillian.
axioms (2)
  • domain assumption The universal operator growth hypothesis asserts at most linear growth of Lanczos coefficients.
    Invoked as the bound that is exceeded.
  • domain assumption Large-central-charge limit of W3 CFTs.
    Stated as the regime in which the descendant module is analyzed.

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