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

TL;DR

This work introduces an infinite class of RG flows between diagonal ${\mathcal{W}}_N(p,q)$ minimal models, of the form ${\mathcal{W}}_N(p,q)\to{\mathcal{W}}_N(p,kp-q)$, governed by adjoint-type deformations that preserve a large non-invertible symmetry $\mathrm{Rep}[SU(N)_{p-N}]$ (and related quotients). The flows are derived via anomaly matching of the preserved Verlinde lines and the monotonicity of the effective central charge $c_{\rm eff}$, yielding a uniform description that is independent of $N$ and extends known Virasoro flows to ${\mathcal{W}}_N$ theories. Specifically, the paper works out detailed examples for ${\mathcal{W}}_3$ and ${\mathcal{W}}_4$ and then generalizes to the full family ${\mathcal{W}}_N$, including special cases where half-integer flows preserve smaller fusion subcategories. The results suggest a deep ${\mathcal{W}}_\infty$-structure underlying truncations of the ${\mathcal{W}}_N$ algebras and point to broad future directions, including numerical tests, RG interfaces, and extensions to non-diagonal invariants and supersymmetric cases.

Abstract

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 ${\mathrm{Rep}}[SU(N)_{p-N}]$ (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.

Figures (2)

• Figure 1: Flows between ${\mathcal{W}}_3(7,q)$ minimal models. Cf. Appendix B for more details and examples for $3\leq q\leq 8$.
• Figure 2: Flows between ${\mathcal{W}}_4(6,q)$ minimal models. The flows in magenta (between top and bottom lines) are the one that preserve only ${\rm Rep}[(SU(4)/\mathbb{Z}_2)_{q-4}]$ category. Cf. Appendix C for more details and detailed discussion $4\leq q\leq 8$