Stability conditions and Lagrangian cobordisms

Abstract

In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category $D\mathcal{F}uk(M)$ of a symplectic manifold $(M,\omega)$ induces a stability condition on the derived Fukaya category of Lagrangian cobordisms $D\mathcal{F}uk(\mathbb{C} \times M)$. We also discuss a relation between stability conditions and the Lagrangian cobordisms group. This provides a general framework in which Haug's result, that the Lagrangian cobordism group of $T^2$ is isomorphic to $K_0(D\mathcal{F}uk(T^2))$, can be understood.