Extending Landau-Ginzburg models to the point

TL;DR

This paper classifies extended topological quantum field theories valued in Landau-Ginzburg models, linking algebraic structures to topological invariants and extending previous constructions through the cobordism hypothesis.

Contribution

It provides a classification of 2-1-0-extended TQFTs with Landau-Ginzburg targets, including conditions for orientability and explicit computations of Serre automorphisms.

Findings

Every polynomial W defines a framed extended TQFT.

W determines an oriented TQFT if its matrix factorisation category is (n-2)-Calabi-Yau.

Constructed TQFTs assign the non-separable Jacobi algebra to a circle.

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.