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 $D\mathcal{F}uk(\mathbb{C} \times M)$. In addition, using stability conditions, we provide general conditions under which the homomorphism $\Theta: \Omega_{Lag}(M)\to K_0(D\mathcal{F}uk(M))$, introduced by Biran and Cornea, is an isomorphism. This yields a better understanding of how stability conditions affect $\Theta$ and it allows us to elucidate Haug's result, that the Lagrangian cobordism group of $T^2$ is isomorphic to $K_0(D\mathcal{F}uk(T^2))$.