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Can one explain to me what is the Hochschild homology of Fukaya category? I mean the definition. You can use the notations of FOOO (Fukaya-Oh-Ono-Ohta) if it helps you to explain easier.

I know what the Fukaya category is but I am very poor when it comes to algebra.

Also please explain what is the corresponding Hochschild homology in B-side?(please include the definition too)

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You can take the Hochschild homology of any algebra or category. For algebras, see the first few sections of Loday's book "Cyclic Homology". There is a simple construction using the standard bar resolution. For categories, there is a similar "bar complex" construction, you can find it for example in Costello's paper on Calabi-Yau categories. For a more abstract approach, see Toën's "The homotopy theory of dg categories..." – Kevin H. Lin Feb 1 '11 at 2:26
(By category I mean linear category or dg category or A-infinity category.) – Kevin H. Lin Feb 1 '11 at 2:32
up vote 11 down vote accepted

As Kevin comments, Hochschild homology and cohomology are defined for any $A_\infty$-category $\mathcal{A}$. That includes Fukaya categories of symplectic manifolds and dg enhancements of the bounded derived category of varieties.

The most concrete definition of Hochschild homology $HH_\ast(\mathcal{A},\mathcal{A})$ is via the cyclic bar complex. One takes the direct sum over all $d\geq 0$ and all sequences of objects $X_0,\dots, X_d$ of the tensor product $$ \hom(X_d,X_0) \otimes \hom (X_{d-1},X_d) \otimes \dots \otimes \hom(X_1,X_2) \otimes \hom(X_0,X_1). $$ You should picture this tensor product not as a linear chain but as circular one; the term $\hom(X_d,X_0)$ is special. In the case where $\mathcal{A}$ is the Fukaya category $\mathcal{F}(M)$ of a symplectic manifold $M$, the $X_i$ are (decorated) Lagrangian submanifolds, and when these are transverse the elements of $\hom(X_i,X_j)$ are linear combinations of intersection points between $X_i$ and $X_j$. So the Hochschild chain complex has a basis given by cyclic sequences of intersection points, one of them marked as special.

The boundary operator is given by taking some sequence of $k\geq 1$ cyclically adjacent terms in the cyclic tensor product and composing them via one of the $A_\infty$-structure maps $\mu^k$ so as to shorten the cyclic sequence by $k-1$. In the Fukaya categorical case, the $\mu^k$ count pseudo-holomorphic $(k+1)$-gons. One does this in all possible ways and sums with hard-to-fathom signs as in Abouzaid's paper 1001.4593 (it would be wonderful if someone can tell me how to make these signs transparent). There is also a chain-lengthening contribution to the complex from the obstruction cochain $\mu^0$.

This concrete description has some real advantages; for instance, as Seidel noticed, there is a geometric description of a homomorphism from Hochschild homology to quantum cohomology

$$ HH_{\ast}(\mathcal{F}(M),\mathcal{F}(M)) \to QH^{\ast}(M) $$

(this for closed $M$) which is expected to be an isomorphism.

For computations, two facts are noteworthy. First, Hochschild homology has Morita-invariance properties. For example, it is unchanged under passing to the category of twisted complexes, which is useful because one can restrict attention to some collection of objects that generate the derived category. Second, it is the derived tensor product of graded bimodules (see Sasha's answer), which means in practical terms that you can compute it using much smaller complexes than the cyclic bar complex.

The conjecture that $HH_{\ast}(\mathcal{F}(M),\mathcal{F}(M))\cong QH^{\ast}(M)$ is consistent with mirror symmetry. In that case, the twisted complexes on $\mathcal{F}(M)$ (technically, the idempotent completion thereof - this doesn't affect $HH_\ast$ either) are quasi-equivalent to a dg-enhanced bounded derived category on the mirror manifold $W$, defined over some non-archimedean Novikov-type field.

Here my understanding is rather feeble, but I think the story is that $HH_\ast$ for this dg category is isomorphic to Hochschild homology of the non-singular variety $W$ (for various equivalent definitions, see Swan's article). This is known to be isomorphic to sheaf cohomology $H^\ast(W, \Omega^\ast_W)$ of the algebraic differential forms, hence to ordinary cohomology of $W$, hence finally to cohomology of $M$. So, if you have an HMS theorem for $M$ and $W$, you at least know that $HH_\ast(\mathcal{F}(M),\mathcal{F}(M))$ is isomorphic to $QH^\ast(M)$.

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Can you explain your notation: what is ${\cal F}(M)$? – John Klein Feb 1 '11 at 13:57
John: thanks for pointing out that oversight. I edited to say that it's the Fukaya category of the symplectic manifold $M$. Since the mirror-symmetric context means we should think of compact Calabi-Yau manifolds, the relevant definition of the Fukaya category is based on the technically complicated constructions of Fukaya-Oh-Ohta-Ono's book (but in special cases simpler options are available). – Tim Perutz Feb 1 '11 at 14:15
likes this answer. – Sheikraisinrollbank Feb 1 '11 at 14:27

To define Hochschild homology of a category $C$ one considers the category of its endofunctors $End(C)$. It has a natural tensor structure given by the composition of functors. Moreover, it has a distinguished object $1_C$ --- the identity functor of $C$. The Hochschild homology is defined as self-Tors of this object: $$ HH_\bullet(C) = Tor_\bullet^{End(C)}(1_C,1_C). $$ In some sense this is a meta-definition. To make it work you need to have some structure on $End(C)$ which would allow you to speak about Tors. Usually, one replaces $End(C)$ by an appropriate faithfully flat subcategory which has such a structure. For example, if $C$ is a dg-category, one considers the category of $C$-bimodules. It is a dg-category as well, each bimodule produces a functor (by tensoring), its natural tensor structure corresponds to the composition of functors, and it has a so-called "diagonal bimodule" corresponding to the identity functor. So, in the above formula you should consider $1_C$ as the diagonal bimodule and $End(C)$ as the category of bimodules over $C$.

The Fukaya category, is defined as derived of a certain $A_\infty$ category with which you can perform the same procedure as with dg-category (consider $A_\infty$-bimodules etc). This gives you the definition of Hochschild homology of a Fukaya category.

On B-side you consider the derived category of coherent sheaves. It doesn't have any specific dg-enhancement, but you can pick up any one you like and again apply the above definition.

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Thanks to both of you, but it is still a complicated object for me! May be I can digest it better later. – Mohammad F. Tehrani Feb 1 '11 at 20:12

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