Curved String Topology and Tangential Fukaya Categories

For manifolds $M$ with a specific rational homotopy type, I study a non-commutative Landau-Ginzburg model whose underlying ring is the differential-graded algebra (dga) $B=C_*(\Omega(M))$, that is chains on the based loop space with Pontryagin product and with potential $W$ in $B$. For $M=\mathbb{C}P^{n_1}\times \mathbb{C}P^{n_2} \times \ldots \mathbb{C}P^{n_k}$ or $S^{n_1} \times S^{n_2} \times \ldots S^{n_k}$, we explain how the field theories we define have a Fukaya category interpretation