On the Algebra of the Infrared with Twisted Masses

The Algebra of the Infrared \cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $\mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector. The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $\mathcal{N}=(2,2)$ quantum field theories than was discussed in \cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential new ingredient is the presence of BPS particles within a fixed vacuum sector. In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass. We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras. Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form. This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period. We demonstrate our formalism explicitly in a particularly instructive example.