Title: An operator algebraic approach to fusion category symmetry on the lattice Show Chinese title

Title: 一种算子代数方法研究格点上的融合范畴对称性

Abstract: 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. Show Chinese abstract

Abstract: 我们通过给出对SymTFT分解的正式解释，提出了一种在热力学极限下(1+1)D晶格上融合范畴对称性的框架。 我们的方法基于对准局部可观测量物理边界子代数的公理化，并应用代数量子场论的思想来推导预期的范畴结构。 我们证明，给定一个准局部代数$A$的物理边界子代数$B$，存在一个由双模作用于$A$上的规范融合范畴$\mathcal{C}$，其融合环通过保持局域性的量子通道作用于准局部代数，使得$B$作为不变算子被恢复。 我们证明，当且仅当其所有对象都有整数维度时，融合范畴可以作为张量积自旋链的对称性，并且当且仅当它具有纤维函子时，它可以在线上作用于张量积自旋链。 我们给出了拓扑对称态的形式定义，并证明了一个Lieb-Schultz-Mattis类型的定理。 利用这一点，我们证明对于任何没有纤维函子的融合范畴$\mathcal{C}$，在任意子链上总是存在无能隙的纯对称态。 最后，我们将我们的框架应用于证明，任何在异常Kramers-Wannier型对偶性下协变的状态必须是无能隙的。