Operator-Algebraic Methods for Categorical Symmetries

In quantum many-body systems and quantum field theories, the notion of symmetry extends beyond groups to include higher categorical structures. In one spatial dimension, finite symmetries of this kind are described by unitary fusion categories. One could ask: how can such symmetries be realised on lattice systems, and which symmetries admit such realisations?

 

In this talk, I will first review fusion categories, together with an operator-algebraic framework for these fusion category symmetries in 1+1d lattice models, proposed by Evans and Jones [arXiv:2507.05185]. I will then discuss a recent result of Bunner and Jones [arXiv:2605.21327], showing that every unitary fusion category can be realised on tensor products of Hilbert spaces, provided infinite-dimensional ancillary Hilbert spaces are allowed.