论文标题
融合类别对称性在1+1维度中的效费
Fermionization of fusion category symmetries in 1+1 dimensions
论文作者
论文摘要
我们讨论了二维拓扑量子场理论（TQFTS）中融合类别对称性的效费。当授权TQFT的对称性通过表示半薄弱的Hopf代数$ H $的表示类别$ \ MATHRM {REP}（H）$描述时，Fermionized tqft具有超级信息类别对称类别$ \ \ m m iathrm {srep}（srep}（srep}（srep}（srep}）） Superalgebra $ \ Mathcal {H}^U $。弱的hopf superalgebra $ \ mathcal {h}^u $不仅取决于$ h $，而且还取决于选择用于效法化的非异常$ \ mathbb {z} _2 $ subgroup的$ \ mathrm {rep}（rep}（rep} h）$。我们通过用$ \ mathrm {srep}（\ Mathcal {h}^u）$对称来明确构建费米子TQFT，来得出$ \ Mathcal {H}^U $的通用公式。当对称性是非反对的时，我们还构建了费米尼裂缝相的晶格哈密顿。作为具体的例子，我们计算有限群对称性，有限规理论的对称性和对称对称性的效率。我们发现，二元性对称性的效费取决于原始融合类别的$ f $ - 符号。上述具体示例的计算表明，我们的融合类别对称性的费米化公式也可以应用于非主观QFTS。
We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}(\mathcal{H}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra $\mathcal{H}^u$. The weak Hopf superalgebra $\mathcal{H}^u$ depends not only on $H$ but also on a choice of a non-anomalous $\mathbb{Z}_2$ subgroup of $\mathrm{Rep}(H)$ that is used for the fermionization. We derive a general formula for $\mathcal{H}^u$ by explicitly constructing fermionic TQFTs with $\mathrm{SRep}(\mathcal{H}^u)$ symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on $F$-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.
