Higher categories of push-pull spans, II: Matrix factorizations

This is the second part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. In the first part we constructed a symmetric monoidal $(\mathrm{\infty },3)$-category $\mathcal{C}\mathcal{R}\mathcal{W}$ of commutative Rozansky-Witten models with the goal of approximating the $3$-category of Kapustin and Rozansky. In this paper we extend work of Brunner, Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we exhibit a functor connecting their $2$-category of matrix factorizations with the homotopy $2$-category of $\mathcal{C}\mathcal{R}\mathcal{W}$, and calculate the associated TFTs.