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

TL;DR

This paper studies the renormalization group flow between $W_3$ minimal models using perturbation and domain wall methods, revealing new phenomena like non-factorization of three-point functions and deriving explicit anomalous $W$-weights.

Contribution

It provides the first detailed analysis of $W_3$ minimal model RG flows, including explicit calculations of anomalous dimensions, mixing coefficients, and conserved currents, with a novel observation of non-factorization phenomena.

Findings

Explicit matrices of anomalous dimensions for different RG invariant sets.

Discovery of violation of holomorphic-antiholomorphic factorization in secondary fields.

Agreement between perturbative and domain wall approaches for the RG flow.

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.