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The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There's a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians. Progress on the significance of this map is in the pipeline, by others, and I'm not going to steal their thunder by discussing it here...

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH^{\ast}$ just in terms of $HH^{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?

ADDED: Kevin asks for evidence for or against $QH^{\ast}\to HH^{\ast}$ being an isomorphism. I don't know any evidence contra. Verifying it for a given $X$ would presumably go in two steps: (i) identify generators for the (triangulated envelope of) $F(X)$, and (ii) show that the map from $SH^{\ast}$ to $HH^{\ast}$ for the full subcategory that they generate is an isomorphism. There's been lots of progress on (i), less on (ii), though the case of toric Fanos has been studied by Fukaya-Oh-Ohta-Ono, and in this case mirror symmetry makes predictions for (i) which I expect will soon be proved. In simply connected disc-cotangent bundles, the zero-section generates, and both $HH^{\ast}$ and $SH^{\ast}$ are isomorphic to loop-space homology, but I don't think it's known that the resulting isomorphism is Seidel's.

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There's a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians.

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. (Edit August 2010: I was careless about homology versus cohomology. I should have said that $HH_{\ast}$ maps to $SH^{\ast}$.) Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH_{\ast}$ just in terms of $HH_{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?

ADDED: Kevin asks for evidence for or against $QH^{\ast}\to HH^{\ast}$ being an isomorphism. I don't know any evidence contra. Verifying it for a given $X$ would presumably go in two steps: (i) identify generators for the (triangulated envelope of) $F(X)$, and (ii) show that the map from $QH^{\ast}$ to $HH^{\ast}$ for the full subcategory that they generate is an isomorphism. There's been lots of progress on (i), less on (ii), though the case of toric Fanos has been studied by Fukaya-Oh-Ohta-Ono, and in this case mirror symmetry makes predictions for (i) which I expect will soon be proved. In simply connected disc-cotangent bundles, the zero-section generates, and both $HH_{\ast}$ for the compact Fukaya category and $SH^{\ast}$ are isomorphic to loop-space homology, but I don't think it's known that the resulting isomorphism is Seidel's.

Added August 2010: Abouzaid (1001.4593) has made major progress in this area.

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). The PSS isomorphism says that $QH^{\ast}(X)$ is canonically isomorphic to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There is a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians. Progress on the significance of this map is in the pipeline, by others, and I'm not going to steal their thunder by discussing it here...

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH^{\ast}$ just in terms of $HH^{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There's a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians. Progress on the significance of this map is in the pipeline, by others, and I'm not going to steal their thunder by discussing it here...

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH^{\ast}$ just in terms of $HH^{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?

ADDED: Kevin asks for evidence for or against $QH^{\ast}\to HH^{\ast}$ being an isomorphism. I don't know any evidence contra. Verifying it for a given $X$ would presumably go in two steps: (i) identify generators for the (triangulated envelope of) $F(X)$, and (ii) show that the map from $SH^{\ast}$ to $HH^{\ast}$ for the full subcategory that they generate is an isomorphism. There's been lots of progress on (i), less on (ii), though the case of toric Fanos has been studied by Fukaya-Oh-Ohta-Ono, and in this case mirror symmetry makes predictions for (i) which I expect will soon be proved. In simply connected disc-cotangent bundles, the zero-section generates, and both $HH^{\ast}$ and $SH^{\ast}$ are isomorphic to loop-space homology, but I don't think it's known that the resulting isomorphism is Seidel's.

The statement that $HF(X,X)$ is isomorphic to $QH(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). The PSS isomorphism says that $QH^*(X)$ is canonically isomorphic to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There is a geometrically-defined map $QH(X) \to HH^*(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians. Progress on the significance of this map is in the pipeline, by others, and I'm not going to steal their thunder by discussing it here...

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^*$ of the wrapped Fukaya category, which involves non-compact Lagrangians. Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH$ just in terms of $HH$ for the full subcategory generated by the thimbles. The latter should be related to $SH$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). The PSS isomorphism says that $QH^{\ast}(X)$ is canonically isomorphic to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.

Now to Hochschild cohomology of the Fukaya category $F(X)$. There is a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.

This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians. Progress on the significance of this map is in the pipeline, by others, and I'm not going to steal their thunder by discussing it here...

There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.

The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.

One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH^{\ast}$ just in terms of $HH^{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".

What could be simpler?