We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either $\mathbb{Z}_2$- or $(\mathbb{Z}_2 \times \mathbb{Q})$-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object $W \in \Bbbk[x_1,\dots,x_n]$ determines a framed extended TQFT. We then compute the Serre automorphisms $S_W$ to show that $W$ determines an oriented extended TQFT if the associated category of matrix factorisations is $(n-2)$-Calabi-Yau. The extended TQFTs we construct from $W$ assign the non-separable Jacobi algebra of $W$ to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on $W=x^{N+1}$ given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis.

中文翻译：

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将 Landau-Ginzburg 模型扩展到点

我们用 Landau-Ginzburg 模型的二分类中的值对框架和定向 2-1-0-extended TQFTs 进行分类，其对象和 1-态射是孤立的奇点和（$\mathbb{Z}_2$- 或 $(\mathbb {Z}_2 \times \mathbb{Q})$-graded) 矩阵分解，分别。为此，我们提出了相关的对称幺半群结构，并发现每个对象 $W \in \Bbbk[x_1,\dots,x_n]$ 确定了一个带框架的扩展 TQFT。然后我们计算 Serre 自同构 $S_W$ 以表明如果矩阵分解的相关类别是 $(n-2)$-Calabi-Yau，$W$ 确定了一个有向扩展的 TQFT。我们从 $W$ 构建的扩展 TQFT 将 $W$ 的不可分离雅可比代数分配给一个圆。这说明了不可分代数如何出现在 2-1-0-extended TQFTs 中，更一般地说，可扩展性问题取决于目标类别的选择。作为另一个应用，我们展示了基于 Khovanov 和 Rozansky 给出的 $W=x^{N+1}$ 的扩展 TQFT 的构造如何可以直接从协同假设推导出来。