We'll describe a method for getting a handle on the closed string sector of the Landau-Ginzburg B-model, i.e., the Hochschild invariants of the 2-periodic dg-categories of matrix factorizations, by relating it to the Hochschild invariants of the total space of the potential. The starting point will be a description of the 2-periodic dg-category of matrix factorizations as a 'Tate-construction' for a homotopy circle action (or better, $B\widehat{\mathbb{G}}_{a}$-action) on the dg-category of perfect complexes on the total space. Then, we'll explain some algebraic tools allowing one to leverage this description along with the well-known understanding of Hochschild invariants of perfect complexes (e.g., the various formalities).