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I've scanned the paper I linked above for a few minutes(I am saying that so people judge what I say critically) and I'd like to expand on the comment that some category of modules over the deformation quantization is a sort of classical limit of the Fukaya category(e.g. a version without pseudo-holomorphic disks). I am sure there have been advances since this paper, since it goes back to 2002 and is a bit loose, and I'd be really interested to hear what they were. First of all, Bressler and Soibelman have the lemma that I was alluding to above that if $A_X$ is the quantized algebra of smooth functions, $HH^*(A_X-mod)= H^*(X)\otimes C((t))$(I guess Bressler et. al. work with complex valued smooth functions.) with the ordinary product. Meanwhile (assuming one has symplectic form that is integral??) the quantum cup product can be defined on the same vector space.The difference is that the product in the first vector space is the ordinary product, while the product in the second one is deformed by p.h.d.'s :).

This can be some kind of closed string version of the classical limit analogy. Now, the observation that is really cool in the paper however, is that if one restricts to the category of what they call holonomic modules, one can get a much more precise version of this analogy(a categorical one). Let Hol(X), be the full subcategory of modules whose support(of M/tM) consists of modules are Lagrangian Submanifolds. In a way not specified by BS :), they claim that the data of a langrangian submanifold and a local system (L,p) can be used to produce a holonomic module in a canonical way. Furthermore, one has the following analogy with the Fukaya category... Let $M_{(L,p)}$ be the module produced above... One has $Ext^*(M_{(L,p)},M_{(L,p)})=H^*(L,p)\otimes C((t))$. The obvious proposition about two Lagrangian's with transversal support is correct too, see Prop 2. on page 12. So again, this category has the right Hom's as vector spaces, but not as an $A(\infty)$ category (the Hom's are again the Hom's in the absence of holomorphic disks).

Bressler and Soibelman then continue with some speculation about algebraic ways to put in the discs, their basic idea being to find a functor $\phi : hol(X) \mapsto hol(X)$ such that $Hom(A,\phi(B)$ agrees as an A-infinity category with the Fukaya category. By the dg category yoga, this should come in the form of a bi-module. They then explain roughly how to do it in the cotangent bundle case. Their vision seems to be that maybe one could have a purely algebraic approach to the Fukaya category via holonomic modules and thereby avoid some of the technical issues with moduli spaces of discs. This also gives an approach to the co-isotropic branes people have been talking about.

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I've scanned the paper I linked above for a few minutes(I am saying that so people judge what I say critically) and I'd like to expand on the comment that some category of modules over the deformation quantization is a sort of classical limit of the Fukaya category(e.g. a version without pseudo-holomorphic disks). I am sure there have been advances since this paper, since it goes back to 2002 and is a bit loose, and I'd be really interested to hear what they were. First of all, Bressler and Soibelman have the lemma that I was alluding to above $HH^*(A_X-mod)= H^*(X)\otimes C((t))$(I guess Bressler et. al. work with complex valued smooth functions.) with the ordinary product. Meanwhile (assuming one has symplectic form that is integral??) the quantum cup product can be defined on the same vector space.The difference is that the product in the first vector space is the ordinary product, while the product in the second one is deformed by p.h.d.'s :).

This can be some kind of closed string version of the classical limit analogy. Now, the observation that is really cool in the paper however, is that if one restricts to the category of what they call holonomic modules, one can get a much more precise version of this analogy(a categorical one). Let Hol(X), be the full subcategory of modules whose support(of M/tM) consists of modules are Lagrangian Submanifolds. In a way not specified by BS :), they claim that the data of a langrangian submanifold and a local system (L,p) can be used to produce a holonomic module in a canonical way. Furthermore, one has the following analogy with the Fukaya category... Let $M_{(L,p)}$ be the module produced above... One has $Ext^*(M_{(L,p)},M_{(L,p)})=H^*(L,p)\otimes C((t))$. The obvious proposition about two Lagrangian's with transversal support is correct too, see Prop 2. on page 12. So again, this category has the right Hom's as vector spaces, but not as an $A(\infty)$ category (the Hom's are again the Hom's in the absence of holomorphic disks).

Bressler and Soibelman then continue with some speculation about algebraic ways to put in the discs, their basic idea being to find a functor $\phi : hol(X) \mapsto hol(X)$ such that $Hom(A,\phi(B)$ agrees as an A-infinity category with the Fukaya category. By the dg category yoga, this should come in the form of a bi-module. They then explain roughly how to do it in the cotangent bundle case. Their vision seems to be that maybe one could have a purely algebraic approach to the Fukaya category via holonomic modules and thereby avoid some of the technical issues with moduli spaces of discs.