Violation of Universal Operator Growth Hypothesis in $\mathcal{W}_3$Conformal Field Theories

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.