Unobstructed Immersed Lagrangian Correspondence and Filtered 𝐴∞ Functor

Abstract

In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered 𝐴∞ categories. We consider arbitrary (compact) symplectic manifolds and their arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered 𝐴∞ category associated with (𝑋, ω) is defined by using Lagrangian Floer theory in such generality, see Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009). The morphism of an unobstructed immersed Weinstein category (from (𝑋₁, ω₁) to (𝑋₂, ω₂)) is, by definition, a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009)). Such a morphism transforms an (immersed) Lagrangian submanifold of (𝑋₁, ω₁) to one of (𝑋₂, ω₂). The key new result proved in this paper shows that this geometric transformation preserves the unobstructedness of the Lagrangian Floer theory. Thus, this paper generalizes earlier results by Wehrheim-Woodward and Mau-Wehrheim-Woodward so that it works in complete generality in the compact case. The main idea of the proofs is based on Lekili-Lipyanskiy's Y diagram and a lemma from homological algebra, together with systematic use of the Yoneda functor. In other words, the proofs are based on a different idea from those that are studied by Bottmann, Mau, Wehrheim, and Woodward, where strip shrinking and figure 8 bubble play the central role.