Lagrangian Intersection Floer Theory

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

“…Let i → 0 be a sequence of rescaling parameters, and let Fuk(Q; i J) be the corresponding Fukaya categories. The homotopy method (see [17,40]) defines A ∞ equivalences of pre-categories…”

“…We now introduce the relevant moduli spaces for the definition of F J . The technique we are describing is originally due to Floer, although it has been extended by Fukaya, Oh, Ohta, and Ono in their work on Lagrangian Floer homology [17].…”

“…Of course, to make this discussion formal, one introduces appropriate Sobolev spaces. We refer the interested reader to the references [17,40], and simply note that regardless of the details, it is easy to see that virdim(M J (p, q)) = virdim(M Jt,s (p, q)) + 1.…”

“…The homotopy method was extended by Fukaya, Oh, Ohta, and Ono in their joint work [17] to produce A ∞ functors; an account in the exact setting appears in [40,Section 10e]. The end result is that, using the analogue of Proposition 5.5 in the presence of more than one Lagrangian, we can conclude that the pre-categories…”

Homological mirror symmetry for Brieskorn–Pham singularities

“…It is an A ∞ -category whose spaces of morphisms are given by Lagrangian intersection Floer complexes, and A ∞ -operations are defined by counting virtual numbers of stable maps of genus zero with Lagrangian boundary conditions [6]. The directed subcategory of B with respect to the order (C 1 , .…”