Von Neumann subfactors and non-invertible symmetries, by Xingyang Yu, Hao Y. Zhang
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SciPost Physics Home Authoring Refereeing Submit a manuscript About Von Neumann subfactors and non-invertible symmetries Xingyang Yu, Hao Y. Zhang SciPost Phys. 19, 154 (2025) · published 15 December 2025 doi: 10.21468/SciPostPhys.19.6.154 pdf BiBTeX RIS Submissions/Reports Abstract We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry $\mathcal{C}$, its module category $\mathcal{M}$, and a gauging labeled by an algebra object $\mathcal{A}$ are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry $\mathcal{C}'$ obtained by gauging $\mathcal{A}$ in $\mathcal{C}$, as well as a reverse gauging back to $\mathcal{C}$. From a given subfactor $N \subset M$, we derive a quiver diagram that encodes the representations of the associated non-invertible symmetry. We show how this framework provides necessary conditions for admissible gaugings, enabling the construction of generalized orbifold groupoids. To illustrate this strategy, we present three examples: Rep$(D_4)$ as a warm-up, the higher-multiplicity case Rep$(A_4)$ with its associated generalized orbifold groupoid and triality symmetry, and Rep$(A_5)$, where $A_5$ is the smallest non-solvable finite group. For applications to gapless systems, we embed these generalized gaugings as global manipulations on the conformal manifolds of $c=1$ CFTs and uncover new self-dualities in the exceptional $SU(2)_1/A_5$ theory. For $\mathcal{C}$-symmetric TQFTs, we use the subfactor-derived quiver diagrams to characterize gapped phases, describe their vacuum structure, and classify the recently proposed particle-soliton degeneracies. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.19.6.154TI - Von Neumann subfactors and non-invertible symmetriesPY - 2025/12/15UR - https://scipost.org/SciPostPhys.19.6.154JF - SciPost PhysicsJA - SciPost Phys.VL - 19IS - 6SP - 154A1 - Yu, XingyangAU - Zhang, Hao Y.AB - We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry $\mathcal{C}$, its module category $\mathcal{M}$, and a gauging labeled by an algebra object $\mathcal{A}$ are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry $\mathcal{C}'$ obtained by gauging $\mathcal{A}$ in $\mathcal{C}$, as well as a reverse gauging back to $\mathcal{C}$. From a given subfactor $N \subset M$, we derive a quiver diagram that encodes the representations of the associated non-invertible symmetry. We show how this framework provides necessary conditions for admissible gaugings, enabling the construction of generalized orbifold groupoids. To illustrate this strategy, we present three examples: Rep$(D_4)$ as a warm-up, the higher-multiplicity case Rep$(A_4)$ with its associated generalized orbifold groupoid and triality symmetry, and Rep$(A_5)$, where $A_5$ is the smallest non-solvable finite group. For applications to gapless systems, we embed these generalized gaugings as global manipulations on the conformal manifolds of $c=1$ CFTs and uncover new self-dualities in the exceptional $SU(2)_1/A_5$ theory. For $\mathcal{C}$-symmetric TQFTs, we use the subfactor-derived quiver diagrams to characterize gapped phases, describe their vacuum structure, and classify the recently proposed particle-soliton degeneracies.ER - × @Article{10.21468/SciPostPhys.19.6.154, title={{Von Neumann subfactors and non-invertible symmetries}}, author={Xingyang Yu and Hao Y. Zhang}, journal={SciPost Phys.}, volume={19}, pages={154}, year={2025}, publisher={SciPost}, doi={10.21468/SciPostPhys.19.6.154}, url={https://scipost.org/10.21468/SciPostPhys.19.6.154},} Ontology / Topics See full Ontology or Topics database. Conformal field theory (CFT) Global symmetries Orbifolds Solitons Topological quantum field theories (TQFT) Authors / Affiliations: mappings to Contributors and Organizations See all Organizations. 1 Xingyang Yu, 2 Hao Y. Zhang 1 Virginia Tech 2 Kavli Institute for the Physics and Mathematics of the Universe [IPMU] Funders for the research work leading to this publication 文部科学省 Monbu-kagaku-shō / Ministry of Education, Culture, Sports, Science and Technology [MEXT] National Science Foundation [NSF] 東京大学 / University of Tokyo [UT]