  We study algebraic structures ($L_\infty$ and $A_\infty$-algebras) introduced
by Gaiotto, Moore and Witten in their recent work devoted to certain
supersymmetric 2-dimensional massive field theories.
  We show that such structures can be systematically produced in any number of
dimensions by using the geometry of secondary polytopes, esp. their
factorization properties. In particular, in 2 dimensions, we produce, out of a
polyhedral "coefficient system", a dg-category $R$ with a semi-orthogonal
decomposition and an $L_\infty$-algebra $\mathfrak g$. We show that $\mathfrak
g$ is quasi-isomorphic to the ordered Hochschild complex of $R$, governing
deformations preserving the semi-orthogonal decomposition.
  This allows us to give a more precise mathematical formulation of the
(conjectural) alternative description of the Fukaya-Seidel category of a Kahler
manifold endowed with a holomorphic Morse function.
