Immersed Lagrangian Floer Theory

By M. Akaho, D. Joyce. 2008

112 citations recorded by the source corpus.

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;\Lambda) for one Lagrangian or HF^*((L,b),(L',b');\Lambda) 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;\Lambda) is a modified, 'quantized' version of cohomology H^*(L;\Lambda) over the Novikov ring \Lambda. In our immersed case, HF^*(L,b;\Lambda) turns out to be a quantized version of the sum of H^*(L;\Lambda) with a \Lambda-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 \Lambda_{CY}. The proofs involve associating a gapped filtered A-infinity algebra over \Lambda or \Lambda_{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,...

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3f7fa51d2274e93fcb164f4af0168fbb80f2db2e

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Citations in the corpus

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, background): “This is not new material; similar or even more general accounts can be found, for example, in [5], [6] and [30].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “A theory of Lagrangian Floer cohomology for immersed Lagrangians has been worked out in [32] using Kuranishi structures, but we will give a definition that is compatible with the definition of [11] using explicit perturbations, with the aim of using it to make computations in Section 1.”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, methodology): “Some references are McDuff and Salamon [52] for symplectic geometry, the author [42] and Harvey and Lawson [27] for Calabi–Yau m-folds and special Lagrangians, Mantegazza [50], Smoczyk [67] and Neves [56] for (Lagrangian) MCF, Fukaya [18, 19], Fukaya, Oh, Ohta and Ono [20] and Seidel [64] for Lagran”
Family Floer cohomology and mirror symmetry (2014, background): “d. (2) The construction of the mirror space from an SYZ perspective also requires understanding the Floer theory of singular ﬁbres. Whenever the ﬁbre is immersed, this Floer theory is well-understood [4]. In the simplest situation, such a ﬁbre is an immersed Lagrangian 2-sphere with a single double ”
On the homological mirror symmetry conjecture for pairs of pants (2010, methodology): “ya category of Pn to include an ‘extra’ object corresponding to the Lagrangian immersion Ln: Sn!Pn. Remark 3.1. A theory of Lagrangian Floer cohomology for immersed Lagrangians has been worked out in [27] using Kuranishi structures, but we will give a denition that is compatible with the denition of”
Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c} (2013, methodology): “ld have been dened and studied by [CP12]. Here we give a generalization to the case of Lagrangian immersion L in a symplectic orbifold X. It is similar to the moduli space of stable discs dened in [AJ10] discussed in Section 3.2, except that we now have interior orbifold marked points, and one has t”