(2+1)维费米子拓扑相的异常与费米子SPT的(3+1)维路径积分态和
Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
AI总结：
本文提出利用费米子拓扑序与对称性分数化数据，构造(3+1)维拓扑不变的组合态和路径积分，从而提供计算(2+1)维费米子拓扑态异常的通用方法，并成功重现时间反演拓扑超导体的$\mathbb{Z}_{16}$异常指标。
摘要：
给定一个(2+1)维费米子拓扑序和全局对称性群 $G$ 的对称性分数化类，我们展示了如何利用精确的组合态和，为费米子 $G$ 对称性保护拓扑态（$G$-FSPT）构造一个(3+1)维拓扑不变的路径积分。这提供了一种计算(2+1)维费米子对称性富集拓扑态中异常的通用方法。等价地，我们的构造为任何允许保持对称性带隙边界的 FSPT 提供了精确的(3+1)维组合态和路径积分，包括自由费米子分类中出现的 AII、AIII、DIII 和 CII 类的(3+1)维拓扑绝缘体和超导体。我们的构造使用费米子拓扑序（由超模张量范畴刻画）和对称性分数化数据，定义了一个包含非平凡演生费米粒子的玻色子理论的(3+1)维路径积分，然后通过对闭合的3-形式 $\mathbb{Z}_2$ 背景规范场求和来凝聚该费米子。此过程涉及许多与费米统计和分数量子数相关的非平凡高形式异常，需要通过依赖于广义自旋结构的 Grassmann 积分来适当抵消。我们展示了该构造如何重现具有 ${\bf T}^2 = (-1)^F$ 的时间反演对称拓扑超导体的 $\mathbb{Z}_{16}$ 异常指标。在数学上，在标准技术假设下，这意味着我们的构造给出了三角剖分4-流形上的组合态和，能够区分所有 $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ 光滑配边类。因此，它包含了 pin$^+$ Dirac 算子的 eta 不变量所编码的拓扑信息，从而给出了一个能够区分奇异光滑结构的态和 TQFT 示例。
英文摘要：
Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $\mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $\mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.
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