RG flows of minimal $\mathcal{W}$-algebra CFTs via non-invertible symmetries

In this letter we study renormalization group (RG) flows between 2d conformal field theories enjoying extended higher-spin $\mathcal{W}$-symmetry. We propose a new class of RG flows between the diagonal minimal models of ${\mathcal{W}}_N$-algebra that take the form ${\mathcal{W}}_N(p,q)\to {\mathcal{W}}_N(p,kp-q)$. These are obtained by matching the anomalies of the non-invertible symmetry $\mathrm{R}\mathrm{e}\mathrm{p}[SU(N{)}_{p-N}]$ (and its discrete quotients) that is preserved by special relevant primary fields. This large non-invertible symmetry includes the familiar ${\mathbb{Z}}_N$ symmetry of the minimal models. Our new flows furnish a significant generalization of the ones recently found in the case of Virasoro algebra, and include all previously known RG flows of ${\mathcal{W}}_N$. They have the remarkable property of being uniform in the rank $N$ of the $\mathcal{W}$-algebra.