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arXiv:2412.21107 · SciPost Physics

Coupled minimal models revisited II: Constraints from permutation symmetry

António Antunes, Connor Behan

Published Apr 23, 2025• 9 citations
arXivHTMLPDF

Abstract

Coupling N N large m m minimal models and flowing to IR fixed points is a systematic way to build new classes of compact unitary 2d CFTs which are likely to be irrational, and potentially have a positive Virasoro twist gap above the vacuum. In this paper, we build on the construction of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], establishing that, for spins less than 10, additional currents transforming in non-trivial irreducible representations of the permutation symmetry S_N S N are not conserved at the IR fixed points. Along the way, we develop a finer understanding of the spectrum of these theories, of the special properties of the N=4 N = 4 case and of non-invertible symmetries that constrain them. We also discuss variations of the original setup of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], some of which can exist for smaller values of the UV central charge.

DOI
10.21468/scipostphys.18.4.132

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Coupling N N large m m minimal models and flowing to IR fixed points is a systematic way to build new classes of compact unitary 2d CFTs which are likely to be irrational, and potentially have a positive Virasoro twist gap above the vacuum. In this paper, we build on the construction of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], establishing that, for spins less than 10, additional currents transforming in non-trivial irreducible representations of the permutation symmetry S_N S N are not conserved at the IR fixed points. Along the way, we develop a finer understanding of the spectrum of these theories, of the special properties of the N=4 N = 4 case and of non-invertible symmetries that constrain them. We also discuss variations of the original setup of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], some of which can exist for smaller values of the UV central charge.
We study coupled unitary Virasoro minimal models in the large rank (m→∞) limit. In large m perturbation theory, we find two nontrivial IR fixed points which exhibit irrational coefficients in several anomalous dimensions and the central charge. For N>4 copies, we show that the IR theory breaks all possible currents that would otherwise enhance the Virasoro algebra, up to spin 10. This provides strong evidence that the IR fixed points are examples of compact, unitary, irrational conformal field theories with the minimal amount of chiral symmetry. We also analyze anomalous dimension matrices for a family of degenerate operators with increasing spin. These display further evidence of irrationality and begin to reveal the form of the leading quantum Regge trajectory.
We study coupled unitary Virasoro minimal models in the large rank (m→∞) limit. In large m perturbation theory, we find two nontrivial IR fixed points which exhibit irrational coefficients in several anomalous dimensions and the central charge. For N>4 copies, we show that the IR theory breaks all possible currents that would otherwise enhance the Virasoro algebra, up to spin 10. This provides strong evidence that the IR fixed points are examples of compact, unitary, irrational conformal field theories with the minimal amount of chiral symmetry. We also analyze anomalous dimension matrices for a family of degenerate operators with increasing spin. These display further evidence of irrationality and begin to reveal the form of the leading quantum Regge trajectory.
In this paper we continue to investigate the lattice models obtained from 2d CFTs via the construction introduced in [arXiv:2112.01563]. On the side of the 2d CFT we consider the cloaking boundary condition relative to a fixed fusion category F of topological line defects. The resulting lattice model realises the topological symmetry F exactly. We compute the state spaces and Boltzmann weights of these lattice model in the example of unitary Virasoro minimal models. We work directly with amplitudes, rather than with normalised correlators, and we provide a careful treatment of the Weyl anomaly factor in terms of the Liouville action. We numerically evaluate the Ising CFT on the torus with one hole and cloaking boundary condition in two channels, and illustrate in this example that the anomaly factors are essential to obtain matching results for the amplitudes. We show that lattice models obtained from Virasoro minimal models at lowest non-trivial cutoff can be exactly mapped to loop models. This provides a first non-trivial check that our lattice models can contain the 2d CFT they were constructed from in their phase diagram, and we propose a condition on the cloaking boundary condition for F under which we expect this to happen in general.
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.
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.