We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe ’t Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to “ungauge” the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

我们研究由无逆拓扑缺陷线生成的 1+1D 量子场论的广义离散对称性。特别是，我们描述了 't Hooft 异常并对由这些对称性稳定的带隙相进行分类，包括新的 1+1D 拓扑相。这些算子的代数不是一个群，而是通过它们的融合环和交叉关系来描述，以代数形式捕获为融合类别。此类数据定义了 2+1D 的 Turaev-Viro/Levin-Wen 模型，而具有该融合类别作为全局对称性的 1+1D 系统定义了边界条件。这类似于在 Dijkgraaf-Witten 理论的边界上测量离散的全局对称性。我们描述了如何在这些边界条件下“测量”融合类别的对称性，并将对称性保持相与对称性破坏相分开。对于 Tambara-Yamagami 范畴及其概括，它们与轨道折叠下的 Kramers-Wannier 式自对偶相关，我们开发了规范理论技术来简化分析。我们提供了一些具有融合类别对称性的 CFT 示例，这些对称性源自 Kramers-Wannier 式对偶性，作为第二部分配套论文的开胃菜。