Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.18667

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

Federico Ambrosino, Tomáš Procházka

Published Jan 26, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2601.18667

Related papers

Minimal Model Renormalization Group Flows: Noninvertible Symmetries and Nonperturbative Description

Federico Ambrosino, Stefano Negro

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.

arXiv:2501.07511•Physical Review Letters•Jun 18, 2025•5 citations

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

Tanaka, Takahilo, Yu Nakayama

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.

arXiv:2407.21353•Jul 31, 2024•0 citations

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

Yu Nakayama, Takahiro Tanaka

A bstract Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models $$ \mathcal{M} $$ M ( kq + I , q ) → $$ \mathcal{M} $$ 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. $$ \mathcal{M} $$ M (10, 3) → $$ \mathcal{M} $$ 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.

Journal of High Energy Physics•Nov 26, 2024•18 citations

Integrable RG flows on topological defect lines in 2D conformal field theories

Thiago Silva Tavares, Madhav Sinha, Linnea Grans-Samuelsson +2 more

A bstract Topological defect lines (TDLs) in two-dimensional conformal field theories (CFTs) are standard examples of generalized symmetries in quantum field theory. Integrable lattice incarnations of these TDLs, such as those provided by spin/anyonic chains, provide a crucial playground to investigate their properties, both analytically and numerically. Here, a family of parameter-dependent integrable lattice models is presented, which realize different TDLs in a given CFT as the parameter is varied. These models are based on the general quantum-inverse scattering construction, and involve inhomogeneities of the spectral parameter. Both defect hamiltonians and (defect) line operators are obtained in closed form. By varying the inhomogeneities, renormalization group flows between different TDLs (such as the Verlinde lines associated with the Virasoro primaries (1 , s ) and ( r, 1) in diagonal minimal CFTs) are then studied using different aspects of the Bethe-ansatz as well as ab-initio numerical techniques. Relationships with the anisotropic Kondo model as well as its non-Hermitian version are briefly discussed.

arXiv:2408.08241•Journal of High Energy Physics•Apr 18, 2025•6 citations

New RG flows between non-Unitary CFTs from exact massless scattering theories

Changrim Ahn, Zoltán Bajnok, Soma Elek

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.

arXiv:2608.24385•Aug 25, 2026•0 citations

Topological defects in conformal field theories and their lattice realizations

Madhav Sinha

Topological defects in two-dimensional conformal field theories (CFTs) are canonical examples of generalized symmetries in quantum field theories. They not only encode key features of the underlying theory, but their study has also enhanced our understanding of renormalization group (RG) flows in theories that contain them. Lattice models exist which discretize the states, energy-momentum tensor, and primary fields of many 2D CFTs, raising the natural question of whether topological defects can be similarly discretized. In this thesis, we demonstrate that in many cases all topological defects of minimal model CFTs can be discretized, enabling analytical and ab-initio numerical computations of their properties. In some of these constructions, we make essential use of the integrable structure of the underlying lattice models. The lattice realization of these operators also permits the study of defect RG flows between CFTs with different topological defects inserted. These topological defects and defect RG flows are realized on the lattice not only for Hermitian but also for non-Hermitian lattice Hamiltonians. Studying topological defects in minimal model CFTs on the lattice has also opened the door to a deeper understanding of topological defects in CFTs where they remain poorly understood.

•Jul 13, 2026•0 citations
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.
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.
A bstract Topological defect lines (TDLs) in two-dimensional conformal field theories (CFTs) are standard examples of generalized symmetries in quantum field theory. Integrable lattice incarnations of these TDLs, such as those provided by spin/anyonic chains, provide a crucial playground to investigate their properties, both analytically and numerically. Here, a family of parameter-dependent integrable lattice models is presented, which realize different TDLs in a given CFT as the parameter is varied. These models are based on the general quantum-inverse scattering construction, and involve inhomogeneities of the spectral parameter. Both defect hamiltonians and (defect) line operators are obtained in closed form. By varying the inhomogeneities, renormalization group flows between different TDLs (such as the Verlinde lines associated with the Virasoro primaries (1 , s ) and ( r, 1) in diagonal minimal CFTs) are then studied using different aspects of the Bethe-ansatz as well as ab-initio numerical techniques. Relationships with the anisotropic Kondo model as well as its non-Hermitian version are briefly discussed.
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.
Topological defects in two-dimensional conformal field theories (CFTs) are canonical examples of generalized symmetries in quantum field theories. They not only encode key features of the underlying theory, but their study has also enhanced our understanding of renormalization group (RG) flows in theories that contain them. Lattice models exist which discretize the states, energy-momentum tensor, and primary fields of many 2D CFTs, raising the natural question of whether topological defects can be similarly discretized. In this thesis, we demonstrate that in many cases all topological defects of minimal model CFTs can be discretized, enabling analytical and ab-initio numerical computations of their properties. In some of these constructions, we make essential use of the integrable structure of the underlying lattice models. The lattice realization of these operators also permits the study of defect RG flows between CFTs with different topological defects inserted. These topological defects and defect RG flows are realized on the lattice not only for Hermitian but also for non-Hermitian lattice Hamiltonians. Studying topological defects in minimal model CFTs on the lattice has also opened the door to a deeper understanding of topological defects in CFTs where they remain poorly understood.