Let (M, !) be a compact symplectic manifold, and L a compact
embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and
Ono [8] construct Lagrangian Floer cohomology, yielding groups
HF(L, b;nov) for one or HF
(L1, b1), (L2, b2);nov
for two
Lagrangians, where b, b1, b2 are choices of bounding cochains, and
exist if and only if L,L1,L2 have unobstructed Floer cohomology.
These are independent of choices up to isomorphism, and have
important invariance properties under Hamiltonian equivalence.
Floer cohomology groups are the morphism groups in the derived
Fukaya category of (M, !), and so are an essential part of the
Homological Mirror Symmetry Conjecture of Kontsevich.
The goal of this paper is to extend [8] to immersed Lagrangians
: L ! M, with transverse self-intersections. In the embedded
case, Floer cohomology HF(L, b;nov) is a modified, ‘quantized’
version of singular homology Hn−(L;nov) over the Novikov ring
nov. In our immersed case, HF(L, b;nov) turns out to be a
quantized version of Hn−(L;nov)
L
(p−,p+)2R nov · (p−, p+),
where R =
(p−, p+) : p−, p+ 2 L, p− 6= p+, (p−) = (p+)
is
a set of two extra generators for each self-intersection point of L,
and (p−, p+) has degree (p−,p+) 2 Z, an index depending on how
L intersects itself at (p−) = (p+).
The theory becomes simpler and more powerful for graded Lagrangians
in Calabi–Yau manifolds, when we can work over a
smaller Novikov ring CY. The proofs involve associating a gapped
filtered A1 algebra over 0
nov or 0
CY to : L ! M, which is independent
of nearly all choices up to canonical homotopy equivalence,
and is built using a series of finite approximations called
AN,0 algebras for N = 0, 1, 2, . . ..