Comments on Summing over Bordisms in TQFT

关于Witten型拓扑量子场论（TQFTs）的Atiyah-Segal公理

On Atiyah-Segal axioms for Witten-type TQFTs

本文在Atiyah-Segal框架下为有质量间隙理论拓扑扭曲构造的Witten型TQFTs提出迹映射配边新定义，证明其幺正性并推测修正配边范畴自洽性。

Orbifolds of n-dimensional defect TQFTs

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincaré duals of triangulated bordisms…

Line and surface defects in Reshetikhin-Turaev TQFT

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Δ$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where …

对3-流形的同调群求和

Summing over homology groups of 3-manifolds

该研究针对三维拓扑量子引力玩具模型，将固定边界的3-流形求和转化为带附加结构的有限生成阿贝尔群求和，给出收敛权重界，论证了与二维TQFT系综平均的等价性及p-群分解性质。

超越同伦的孤子对称性：来自配边的可逆性与来自拓扑量子场论的非可逆性

Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory

本文以4维 $\mathbb{C}P^1$ 模型为例，证明hopfion的 $U(1)$ 对称性并不存在，其守恒律实为3维自旋TQFT生成的非可逆对称性，可逆部分仅为 $\mathbb{Z}_2$ 自旋配边不变量，从而提出超越同伦群的孤子对称性统一描述。