Extending Landau-Ginzburg models to the point

Abstract

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.