Immersed Lagrangian Floer Theory

Abstract

Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and only if L,L' have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M,w), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend all this to immersed Lagrangians L in M with immersion i : L --> M, with transverse self-intersections. In the embedded case, Floer cohomology HF^*(L,b;Λ) is a modified, 'quantized' version of cohomology H^*(L;Λ) over the Novikov ring Λ. In our immersed case, HF^*(L,b;Λ) turns out to be a quantized version of the sum of H^*(L;Λ) with a Λ-module spanned by pairs (p,q) for p,q distinct points of L with i(p)=i(q) in M. The theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring Λ_{CY}. The proofs involve associating a gapped filtered A-infinity algebra over Λor Λ_{CY} to i : L --> M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A_{N,0} algebras for N=0,1,2,...