# Orlov equivalence between Fukaya categories

In his famous paper https://arxiv.org/abs/math/0503632, Orlov proves the following theorem (for simplicity, let's just focus on the Calabi-Yau case)

Theorem(Orlov): Suppose that $$W: \mathbb{A}^d \rightarrow \mathbb{A}$$ is a homogeneous polynomial of degree $$d$$ defined over the complex affine space. Then there is an equivalence between the triangulated category of graded matrix factorizations $$DGrB(W)$$ and the bounded derived category of coherent sheaves on the Calabi-Yau hypersurface $$D^{b}(X)$$, where $$X$$ is the projective variety in $$\mathbb{P}^{d-1}$$ defined by $$W$$.

Such a statement admits generalizations, just to list a few: replace homogeneous by weighted-homogeneous, replace the source $$\mathbb{A}^d$$ by quasi-projective schemes/stacks, replace the target $$\mathbb{A}$$ by locally free sheaves, etc. Actually the above theorem also holds for the dg-enhancement of the two categories, by another paper of Lunts-Orlov: https://arxiv.org/abs/0908.4187. Just mention one more thing: the Fourier-Mukai type theory is also developed for matrix factorizations, see https://arxiv.org/abs/1105.3177 for instance.

The FJRW theory develops a similar closed-string result at the A-side. This nice survey paper by Chiodo-Ruan could be served as an introduction:https://arxiv.org/abs/1307.0939. As you may guess, this question is about the possibility of extending the Orlov correspondence for Fukaya categories. A sample of the conjectural picture could be following:

Let $$(X, \omega)$$ be a closed monotone symplectic manifold, with $$c_1(X) = [\omega]$$. Suppose that $$s$$ is a smooth section of $$K_{X}^{-1}$$ which cuts out a smooth symplectic Calabi-Yau hypersurface. Meanwhile, the section $$s$$ defines a $$\mathbb{C}$$-valued smooth function on the total space $$K_X$$. Then there should be an equivalence $$D^{\pi}Fuk(X,\omega) \cong "DFS^{\pi}(K_X, s)"$$ Certainly one can replace $$K_X^{-1}$$ by smooth complex vector bundles whose determinant equals $$K_X^{-1}$$, or replace manifolds by orbifolds if the Fukaya categories could be defined.

I am quite surprised that my searches show that such a problem hasn't been studied seriously. The only relevant symplectic paper is by Abouzaid-Smith https://arxiv.org/abs/1504.01230. This paper contains some discussions of the Orlov functor for a symplectic LG model. Roughly speaking, one can construct an admissible Lagrangian in the total space of an LG model by parallel transporting a Lagrangian in the fiber along "a circle with one point pushed to infinity".

The interesting feature about the above conjectural picture is the jump of dimension. Moreover, as objects in the category of matrix factorizations could be identified with curved complex of coherent sheaves, the Lagrangians in the Fukaya-Seidel type category should bound holomorphic discs. Moreover, as $$D^{\pi}Fuk(X,\omega)$$ is defined over the Novikov field, the right hand side should also be thought as a deformation induced by partial compactification. The Fourier-Mukai kernel results suggest that there might be a similar picture for generalizations of Lagrangian correspondences.

So my questions: was there any relevant ideas from the symplectic side? Would one expect a general approach? (Of course one can use existed techniques to check examples like Calabi-Yau hypersurfaces in projective spaces, but even for these examples one might run into the non-existing generating criterion for LG models)

• Relevant discussions can be found in Section 7 of Abouzaid-Auroux-Katzarkov: arxiv.org/abs/1205.0053, where it is argued that the Fukaya category of a nearly tropical hypersurface $H$ is related to a partially wrapped Fukaya category of the ambient LG model. May 18 '19 at 19:12