“Fourier-Mukai” functors for Fukaya categories?

I just skimmed a bit of this fresh-off-the-press paper on homological mirror symmetry for general type varieties.

One thing that intrigued me was statement (ii) of Conjecture 3.3. It suggests that, just as there are Fourier-Mukai functors $DCoh(X) \to DCoh(Y)$ associated to objects of $DCoh(X \times Y)$, there are also "Fourier-Mukai" functors $DFuk(M) \to DFuk(N)$ associated to objects of $DFuk(M \times N)$. How does this work, or how might this work? The paper does not seem to explain it.

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I can't speak for these authors, but what I understand by a "Fourier-Mukai" transform between Fukaya categories is the functor between extended Fukaya categories associated with a Lagrangian correspondence. I expect these will appear a good deal in the next few years, in symplectic topology and its applications to low-dim topology and mirror symmetry (cf. these papers of Auroux and Abouzaid-Smith).

A Lagrangian correspondence from $X$ to $Y$ is an embedded Lagrangian submanifold of $X\times Y$ with symplectic form $(-\omega_X)\oplus\omega_Y$. One can take the graph of a symplectomorphism, for instance, or the vanishing moment-map locus $\mu^{-1}(0)$ as a correspondence from a Hamiltonian $G$-manifold $M$ to the quotient $\mu^{-1}(0)/G$. According to Wehrheim-Woodward, a generalised Lagrangian in $X$ is a sequence of symplectic manifolds $X_0=pt., X_1, X_2,\dots,X_d=X$ and Lagrangian correspondences $L_{i,i+1}$ from $X_i$ to $X_{i+1}$. Generalised Lagrangians (subject to the usual sorts of restrictions and decorations) form objects in the extended Fukaya category $F^{\sharp}(X)$, whose $A_\infty$-structure is under construction by Ma'u-Wehrheim-Woodward. If it happens that two adjacent Lagrangian correspondences have a smooth, embedded composition, say $L' = L_{i+1,i+2}\circ L_{i,i+1}$, then deleting $X_{i+1}$ from the chain and substituting $L'$ for its two factors results in an isomorphic object; see arxiv:0905.1368.

The geometric mechanism behind this is the idea of "pseudo-holomorphic quilts", see arXiv:0905.1369, arxiv:math.SG.0606061.

Whilst $F^\sharp(X)$ seems even less tractable than the Fukaya category $F(X)$, we expect that in many cases the embedding $F(X)\to F^\sharp(X)$ induces a quasi-isomorphism of module categories. A Lagrangian correspondence from $X$ to $Y$ induces a "Fourier-Mukai" $A_\infty$-functor between extended Fukaya categories. Even better, we expect that in this way one gets an $A_\infty$-functor $F(X_-\times Y)\to \hom(F^{\sharp}(X),F^{\sharp}(Y))$.

Interpolated paragraph, in response to Kevin's query: The definition of the $A_\infty$-functor associated with a correspondence $C$ from $X$ to $Y$ is very simple, at least on objects. An object of $F^\sharp(X)$ is a chain of Lagrangian correspondences, beginning at the 1-point manifold and ending at $X$. We just tack $C$ to the end of this sequence. The clever thing about this formalism is that when there's a nice geometric way to pass Lagrangians through a correspondence (for instance taking the preimage of $L\subset \mu^{-1}(0)/G$ in $\mu^{-1}(0)\subset M$) the geometric and formal approaches give quasi-isomorphic objects in $F^\sharp(Y)$. (Usually you can't pass a Lagrangian submanifold through a correspondence without making a terrible mess - hence the formal approach.)

For me, all this is exciting because we can at last compute Floer cohomology using its naturality properties, rather than by direct attack on the equations.

[In the new paper that inspired your question, the authors suggest that their F-M kernels should be coisotropic branes in the sense of Kapustin-Orlov. Floer cohomology for such branes is supposed to be some weird mixture of pseudo-holomorphic discs and Dolbeault cohomology over a holomorphic foliation - but there is no concrete proposal on the table and for the moment this is just an intriguing idea. For the purposes of homological mirror symmetry, idempotent endomorphisms in the Fukaya category apparently provide the enlargement that K-O observed was necessary.]

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Given a Lagrangian $L \subset X \times Y$ corresponding to the graph of a symplectomorphism $X \to Y$, I guess it is clear what the corresponding functor $F(X) \to F(Y)$ should be. Moreover it should be clear that it is an equivalence of categories. However, given a general Lagrangian $L \subset X \times Y$, I don't really have any idea of what the corresponding functor $F(X) \to F(Y)$ should look like. Could you explain how this is supposed to work for general $L$, either intuitively or in some simple examples? – Kevin H. Lin Apr 2 '10 at 21:25
I mean, at least intuitively or in good situations, am I just taking the inverse image of a Lagrangian in $X$ along the first projection, intersecting that with $L$, and then taking the image of that along the second projection? – Kevin H. Lin Apr 2 '10 at 21:26
I replaced my earlier comments with a new paragraph in the text. – Tim Perutz Apr 3 '10 at 22:00
@Kevin- Yes, morally you are taking inverse image, intersecting and projecting. If everything happens to be transverse, then this is really true. Otherwise, as Tim points out, you essentially have to treat the correspondence as something formal. – Ben Webster Apr 3 '10 at 23:55