Stability Conditions and Lagrangian Cobordisms

TL;DR

This paper explores how stability conditions influence Lagrangian cobordisms and their categories, establishing new links between cobordism groups and derived Fukaya categories, especially for the torus.

Contribution

It demonstrates that stability conditions on the Fukaya category induce similar conditions on Lagrangian cobordisms and clarifies the relationship between cobordism groups and K-theory.

Findings

Stability conditions induce stability conditions on Lagrangian cobordism categories.

Conditions under which the homomorphism $ heta$ is an isomorphism are provided.

The Lagrangian cobordism group of $T^2$ is shown to be isomorphic to $K_0(D ext{Fuk}(T^2))$.

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 $K_0(D\mathcal{F}uk(T^2))$.