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Infinitely many new renormalization group flows between Virasoro minimal models from non-invertible symmetries

Takahilo Tanaka, Yu Nakayama

TL;DR

The paper uses non-invertible (categorical) symmetries to propose infinitely many RG flows between Virasoro minimal models M(kq+I, q) and M(kq−I, q), driven by the φ_{(1,2k+1)} perturbation. It shows these flows preserve a modular tensor category with SU(2)_{q−2} fusion rules, and that invariants like quantum dimensions and defect-space spins constrain possible trajectories, providing a classification framework that recovers and extends known flows. Through explicit examples for q=3,4,5 (and special q=2 cases), the authors illustrate anomaly matching and duality-defect structures (Tambara-Yamagami) that distinguish distinct flows and support completeness under repeated application. The work suggests a robust map for RG flows in 2D CFTs with non-invertible symmetries, with potential implications for integrable structures and Lagrangian realizations in minimal models.

Abstract

Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models $\mathcal{M}(kq + I, q) \to\mathcal{M}(kq-I, q)$ induced by $φ_{(1,2k+1)}$. They vastly generalize the previously proposed ones $k=I=1$ by Zamolodchikov, $k=1, I>1$ by Ahn and Lässig, and $k=2$ by Dorey et al. All the other $\mathbb{Z}_2$ preserving renormalization group flows sporadically known in the literature (e.g. $\mathcal{M}(10,3) \to \mathcal{M}(8,3)$ studied by Klebanov et al) fall into our proposal (e.g. $k=3, I=1$). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the $SU(2)_{q-2}$ fusion ring.

Infinitely many new renormalization group flows between Virasoro minimal models from non-invertible symmetries

TL;DR

The paper uses non-invertible (categorical) symmetries to propose infinitely many RG flows between Virasoro minimal models M(kq+I, q) and M(kq−I, q), driven by the φ_{(1,2k+1)} perturbation. It shows these flows preserve a modular tensor category with SU(2)_{q−2} fusion rules, and that invariants like quantum dimensions and defect-space spins constrain possible trajectories, providing a classification framework that recovers and extends known flows. Through explicit examples for q=3,4,5 (and special q=2 cases), the authors illustrate anomaly matching and duality-defect structures (Tambara-Yamagami) that distinguish distinct flows and support completeness under repeated application. The work suggests a robust map for RG flows in 2D CFTs with non-invertible symmetries, with potential implications for integrable structures and Lagrangian realizations in minimal models.

Abstract

Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models induced by . They vastly generalize the previously proposed ones by Zamolodchikov, by Ahn and Lässig, and by Dorey et al. All the other preserving renormalization group flows sporadically known in the literature (e.g. studied by Klebanov et al) fall into our proposal (e.g. ). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the fusion ring.
Paper Structure (14 sections, 27 equations, 2 figures, 6 tables)

This paper contains 14 sections, 27 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Virasoro minimal models
  3. Non-invertible symmetries and topological defect lines
  4. Invertible TEZT symmetry
  5. Renormalization group invariants
  6. New renormalization group flows
  7. Renormalization group flow TEXT
  8. Half-integer TEXT
  9. Some examples
  10. TEXT and anomaly matching
  11. TEXT and duality defects
  12. TEXT
  13. Flows to TEXT
  14. Discussions

Figures (2)

  • Figure 1: The Kac table and its fundamental domain $\Gamma$ of $\mathcal{M}(p, q)$.
  • Figure 2: $\mathcal{M}(p, 4)$ are classified in terms of the spin contents and the quantum dimensions of the duality defect $\mathcal{N}$. They constrain the renormalization group flow like this figure. Dotted arrows are possible in the half-integer $k$ flow which does not preserve the duality defect lines.