# Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon.

Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of "classical" data -- the Poisson algebra of functions $C^\infty(M)$ and the cohomology algebra (or rather, Frobenius algebra) $H^\ast(M; \mathbb{C})$ respectively.

Are these two things related somehow?

I am interested in both mathematical and physical answers.

I apologize if this question is naive. I feel like, with a proper understanding of the physics, the answer to this question is probably obviously "yes" or obviously "no". Unfortunately, I don't have a good understanding of the physics.

Edit: From the looks of the discussion below, deformation quantization is perhaps more directly related to the Fukaya category. I welcome any additional remarks on the Fukaya category.

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I think HH* of maybe the category of perfect modules over the deformation quantization should also be H*(X)((t)) but as an algebra so I feel like the word "classical limit" should appear here somewhere... In general, I believe that if X is a Poisson manifold, then the HH* of the category of deformed modules should be the Poisson cohomology of X tensored with R((t)). This is all somewhat speculative as I've never seen it written anywhere nor can I really explain an honest proof of these statements but maybe it's in the right direction. –  Daniel Pomerleano Sep 1 '10 at 22:22
This paper by Bressler and Soibelman arxiv.org/abs/hep-th/0202128 has all the words in the title so I bet it has something to say about this question... haven't read it though. –  Daniel Pomerleano Sep 1 '10 at 22:47
The question has to do with pseudo-holomorphic spheres. It would be very interesting and extraordinary if these were related to deformation quantization. I don't know any evidence that they are related. (Cotangent bundles, the subject of Nadler's theorem that "microlocal branes are constructible sheaves", do not contain holomorphic spheres.) –  Tim Perutz Sep 2 '10 at 5:30

## 2 Answers

Yep. The connection I have in mind comes from recent work of Gukov & Witten: "Branes and Quantization". If you've got a symplectic manifold $M$, you can sometimes make the Kontsevich deformation of $C^\infty(M)$ act on the vector space of $A$-model states which describe strings in complexification $Y$ of $M$ which begin on the canonical co-isotropic brane in $Y$ and end on $M$ (which you can think of as a Lagrangian brane in $Y$).

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I'm not sure what a complexification of a symplectic manifold is (I guess it's a complex manifold with some kind of involution such that the fixed point set of the involution is the symplectic manifold?), but it sounds like you're saying that the deformation of $C^\infty(M)$ acts on something like $HF^\ast(M \hookrightarrow M \times M, M \hookrightarrow M \times M)$, which is supposed to be quantum cohomology... Am I on the right track? –  Kevin H. Lin Sep 1 '10 at 21:57
Your definition of complexification is the one G&W use. I don't understand $HF^*$ well enough to render an opinion on the latter. The connection I had in mind is a little more tenuous: small quantum cohomology is the space of states of the closed string $A$-model. –  userN Sep 1 '10 at 23:23
To flesh out what I'm thinking: I think that strings which begin and end on the diagonal $\Delta$ in $M\times M$ are supposed to be described by the Hom of $\Delta$ to itself in the Fukaya category of $M\times M$. Correct? And then this Hom is $HF^\ast(\Delta, \Delta)$, which is supposed to be quantum cohomology of $M$. Moreover, $HF^\ast(\Delta,\Delta)$ is supposed to be $HH^\ast$ of the Fukaya category of $M$, which is the space of states for the closed string A-model for $M$. See: mathoverflow.net/questions/11081/… –  Kevin H. Lin Sep 2 '10 at 0:22
The main problem with deformation quantization is that it does not detect global phenomena (like e.g. instantons in the physics langage), while quantum cohomology does. Namely, if you look at the Poisson $\sigma$-model (the "physical theory" that produces deformation quantization): the way Cattaneo and Felder work with it to get some deformation quantization results (even in the presence of branes) is by working perturbatively around a constant solution of the Master Equation. For a few reference see arxiv.org/abs/math/0309180 and arxiv.org/abs/math/0102108 –  DamienC Sep 2 '10 at 6:22
No one has successfully demonstrated HH* = QH* in a juicy example. Usually, you don't know Fuk (e.g., the two-torus is not exact, but QH* is not interesting). The link bet. def. quantization and Fuk has been appealing for years, but results are mild. (Kapustin-Witten showed D-modules are A-branes, of which the Fuk is the subcategory of Lag. objects, but when your symplectic manifold is not holomorphic-symplectic, you're screwed. This thread circles around the research program of Tsygan et al, which relates to Bressler-Soibelman. Here's a reference: front.math.ucdavis.edu/0905.0290 –  Eric Zaslow Sep 2 '10 at 17:01

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 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 pseudo-holomorphic discs.

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. Bressler and Soibelman 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 know that sometimes $HF^\ast(L,L) \cong H^\ast(L)$. But I think this fails sometimes, right? I am not an expert of Floer theory / Fukaya category... –  Kevin H. Lin Sep 2 '10 at 14:45
Yeah this fails in general... though in the relatively spin case there is always a spectral sequence... the authors make several simplifying assumptions like m_0=0 and "all the assumptions made in FOOO". Anyways, that paper is really just a sketch of an idea. It's also possible that I've missed the point. Maybe some experts will chime in with something more state of the art. –  Daniel Pomerleano Sep 2 '10 at 15:42
Anyway, thanks a lot for your interesting remarks and synopsis of Bressler-Soibelman. –  Kevin H. Lin Sep 2 '10 at 15:43
The abstract of the last paper from last year by Tsygan mentioned by Damien opens up with "we develop further the ideas of Bressler- Soibelman" so maybe that is the state of the art thing to read if you are motivated enough. –  Daniel Pomerleano Sep 2 '10 at 16:03