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arXiv:2407.21353

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

Tanaka, Takahilo, Yu Nakayama

Published Jul 31, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2407.21353

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Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models M(kq+I,q)→M(kq−I,q)\mathcal{M}(kq + I, q) \to\mathcal{M}(kq-I, q)M(kq+I,q)→M(kq−I,q) induced by ϕ(1,2k+1)ϕ_{(1,2k+1)}ϕ(1,2k+1)​. They vastly generalize the previously proposed ones k=I=1k=I=1k=I=1 by Zamolodchikov, k=1,I>1k=1, I>1k=1,I>1 by Ahn and Lässig, and k=2k=2k=2 by Dorey et al. All the other Z2\mathbb{Z}_2Z2​ preserving renormalization group flows sporadically known in the literature (e.g. M(10,3)→M(8,3)\mathcal{M}(10,3) \to \mathcal{M}(8,3)M(10,3)→M(8,3) studied by Klebanov et al) fall into our proposal (e.g. k=3,I=1k=3, I=1k=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−2SU(2)_{q-2}SU(2)q−2​ fusion ring.
A bstract Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models M \mathcal{M} M M ( kq + I , q ) → M \mathcal{M} M M ( kq – I , q ) induced by ϕ (1,2 k +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 ℤ 2 preserving renormalization group flows sporadically known in the literature (e.g. M \mathcal{M} M M (10, 3) → M \mathcal{M} M 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.
RG flows of minimal W\mathcal WW-algebra CFTs via non-invertible symmetries
In this letter we study renormalization group (RG) flows between 2d conformal field theories enjoying extended higher-spin W\mathcal{W}W-symmetry. We propose a new class of RG flows between the diagonal minimal models of WN\mathcal{W}_NWN​-algebra that take the form WN(p,q)→WN(p,kp−q)\mathcal{W}_N(p,q)\to\mathcal{W}_N(p,kp-q)WN​(p,q)→WN​(p,kp−q). These are obtained by matching the anomalies of the non-invertible symmetry Rep[SU(N)p−N]{\mathrm{Rep}}[SU(N)_{p-N}]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 ZN\mathbb{Z}_NZN​ 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 WN\mathcal{W}_NWN​. They have the remarkable property of being uniform in the rank NNN of the W\mathcal{W}W-algebra.
We construct complete solutions of the massless S-matrix bootstrap equations describing a new infinite family of integrable RG flows from the higher-fusion-level minimal models M(3,2n+3;n) to the non-unitary Virasoro minimal models M(2,2n+3), including a flow from M(3,5) to the Yang--Lee CFT M(2,5) when n=1. We provide strong evidence for this identification by matching conformal perturbation theory around the UV fixed points with the small-volume expansion of the corresponding thermodynamic Bethe ansatz equations. We further show that a family of non-invertible Verlinde defect lines is preserved along the flows.
In this Letter we continue the investigation of RG flows between Virasoro minimal models of two-dimensional conformal field theories that are protected by noninvertible symmetries. RG flows leaving unbroken a subcategory of noninvertible symmetries are associated with anomaly matching conditions that we employ systematically to map the space of flows between minimal models beyond the Z_{2}-symmetric proposed recently in the literature. We introduce a family of nonlinear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including nonintegrable cases, such as the recently proposed M(kq+I,q)→M(kq-I,q). Our family of NLIEs encompasses and generalizes the integrable flows known in the literature: ϕ_{(1,3)}, ϕ_{(1,5)}, ϕ_{(1,2)} and ϕ_{(2,1)}. This work uncovers a new interplay between exact solvability and noninvertible symmetries. Furthermore, our nonperturbative description provides a nontrivial test for all the flows conjectured by anomaly matching conditions, but so far not observed by other means.
We study the role of generalized symmetry in massless renormalization group flows or Higgs transitions. In particular, we revisit the massless renormalization group flows in unitary minimal models, M(p,p+1)→M(p−1,p)‾\mathbf{M}(p,p+1) \rightarrow \overline{\mathbf{M}(p-1,p)}M(p,p+1)→M(p−1,p)​ preserving the fusion ring symmetry FR(SU(2)p−2)⊂M(p,p+1)\text{FR} (SU(2)_{p-2})\subset \mathbf{M}(p,p+1)FR(SU(2)p−2​)⊂M(p,p+1) where ppp is an integer satisfying p>3p>3p>3. In this series of flows, we demonstrate that the unbroken fusion ring symmetry FR(SU(2)p−2)\text{FR} (SU(2)_{p-2})FR(SU(2)p−2​) eliminates all relevant perturbations in the M(p−1,p)‾\overline{\mathbf{M}(p-1,p)}M(p−1,p)​ model. Hence, the infrared theory M(p−1,p)‾\overline{\mathbf{M}(p-1,p)}M(p−1,p)​ is stable at the level of the scaling analysis and can be interpreted as a (weak-)symmetry-enforced gapless phase in contemporary theoretical physics. Phenomenologically, by the folding trick, the unbroken fusion ring symmetry corresponds to a (half-)integer spin nonsimple current, a variant of the Cooper pair involving nonabelian anyons generated from the coset or level-rank duality structures. Moreover, we demonstrate that the structure of (half-)integer spin nonsimple current plays a fundamental role in causing the resonance effect of relevant and dangerously irrelevant perturbations. This resonance effect may result in the cascade of phase transitions (or the system flows to unexplored fixed points), and the symmetry can be a stopper of such unconventional flows.
We classify connected étale algebras AAA's in multiplicity-free modular fusion categories B\mathcal BB's with rank(B)≤9\text{rank}(\mathcal B)\le9rank(B)≤9. We also identify categories BA\mathcal B_ABA​'s of right AAA-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous B\mathcal BB-symmetry breaking.