Classification of (2+1)D invertible fermionic topological phases with symmetry
Classification of (2+1)D invertible fermionic topological phases with symmetry
复制标题
具有对称性的 (2 1)D 可逆费米子拓扑相的分类
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
N. Manjunath
中科院分区：
文献类型：
--
作者：
M. Barkeshli;Yu;Po;N. Manjunath
我们为一般费米子对称群 $G_f$ 和手性中心电荷 $c_-$ 的一般值提供了在两个空间维度上具有对称性的相互作用费米子的可逆拓扑相的分类。这里 $G_f$ 是玻色子对称群 $G_b$ 通过费米子宇称 $(-1)^F$ 的中心扩展，由 mathcal{H}^2(G_b, mathbb{Z}_2)$ 中的第二个上同调类 $[omega_2] 指定。我们的方法通过测量费米子奇偶性并对所得的 $G_b$ 对称性丰富的拓扑顺序进行分类，同时跟踪某些附加数据和约束。我们通过两个视角进行分析，使用 $G$ 交叉编织张量类别和 Spin$(2c_-)_1$ Chern-Simons 理论与背景 $G$ 规范场相结合。这些结果提供了一种根据一组具体的数据和一致性方程来表征和分类可逆费米子拓扑相的方法，这比使用配边理论和谱序列的更抽象的方法更物理透明且计算更简单。我们的结果还概括了 Wang 和 Gu 近期对费米子对称保护拓扑相的分类，并提供了一种不同的方法，这些拓扑相具有手性中心电荷 $c_- = 0$。我们展示了拓扑绝缘体和超导体的 10 倍分类如何适合我们的方案，以及由于 $c_-$ 和 $G_f$ 的某些选择而产生的一般非微扰约束。从数学上讲，我们的结果还提出了具有 $G_f$ 对称性的 (2+1)D 可逆拓扑量子场论变形类的显式一般参数化。
We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups $G_f$ and general values of the chiral central charge $c_-$. Here $G_f$ is a central extension of a bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, specified by a second cohomology class $[omega_2] in mathcal{H}^2(G_b, mathbb{Z}_2)$. Our approach proceeds by gauging fermion parity and classifying the resulting $G_b$ symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using $G$-crossed braided tensor categories and Spin$(2c_-)_1$ Chern-Simons theory coupled to a background $G$ gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge $c_- = 0$. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of $c_-$ and $G_f$. Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with $G_f$ symmetry.