An operator algebraic approach to fusion category symmetry on the lattice

We propose a framework for fusion category symmetry on the (1+1)D lattice in the thermodynamic limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra $B$ of a quasi-local algebra $A$, there is a canonical fusion category $\mathcal{C}$ that acts on $A$ by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that $B$ is recovered as the invariant operators. We show that a fusion category can be realized as symmetries of a tensor product spin chain if and only if all of its objects have integer dimensions, and that it admits an on-site action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove a Lieb-Schultz-Mattis type theorem. Using this, we show that for any fusion category $\mathcal{C}$ with no fiber functor there always exists gapless pure symmetric states on an anyon chain. Finally, we apply our framework to show that any state covariant under an anomalous Kramers-Wannier type duality must be gapless.