RG flow between $W_3$ minimal models by perturbation and domain wall approaches

TL;DR

The paper analyzes the RG flow between neighboring $W_3$ minimal models $A_2^{(p)}$ and $A_2^{(p-1)}$ initiated by the relevant operator $\varphi(1212)$. It constructs three RG-invariant field sets, computes OPE structure constants, and diagonalizes the resulting anomalous-dimension matrices to obtain explicit UV→IR mixing patterns, including a detailed second-class mixing that is independently reproduced by an RG domain-wall approach. The authors introduce anomalous $W$-weights and derive their leading behavior in terms of structure constants, with results for the first two classes fully explicit and the third class explored with degeneracies that limit diagonalization. They also observe holomorphic–anti-holomorphic factorization violation in correlators involving descendants and establish a domain-wall check that corroborates the perturbative analysis, reinforcing the connection between current-algebra structure and RG flow in higher-rank $W$-algebras. The work opens pathways to general $W_n$ flows and potential holographic interpretations, while highlighting challenges posed by degeneracies and descendant sectors.

Abstract

We explore the RG flow between neighboring minimal CFT models with $W_3$ symmetry. After computing several classes of OPE structure constants we were able to find the matrices of anomalous dimensions for three classes of RG invariant sets of local fields. Each set from the first class consists of a single primary field, the second one of three primaries, while sets in the third class contain six primary and four secondary fields. We diagonalize their matrices of anomalous dimensions and establish the explicit maps between UV and IR fields (mixing coefficients). While investigating the three point functions of secondary fields we have encountered an interesting phenomenon, namely violation of holomorphic anti-holomorphic factorization property, something that does not happen in ordinary minimal models with Virasoro symmetry solely. Furthermore, the perturbation under consideration preserves a non-trivial subgroup of $W$ transformations. We have derived the corresponding conserved current explicitly. We used this current to define a notion of anomalous $W$-weights in perturbed theory: the analog for matrix of anomalous dimensions. For RG invariant sets with primary fields only we have derived a formula for this quantity in terms of structure constants. This allowed us to compute anomalous $W$-weights for the first and second classes explicitly. The same RG flow we investigate also with the domain wall approach for the second RG invariant class and find complete agreement with the perturbative approach.