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I would like to know whether there is a specific relationship between the deformation quantizations of the Poisson algebras of the functions on a symplectic manifold, say $M$, and of the functions on an open subset of $M$, in other words an openly embedded $N\to M$. I'm only starting to get familiar with the literature on deformation quantization and I've yet to see any reference address this question head on. Comments contrasting the symplectic vs. Poisson and finite vs. infinite dimensional cases are also very welcome.

The larger motivation behind this question is to the desire to see deformation quantization as some kind of functor. I know that in general deformations are not unique, but drawing inspiration from the example of the association of Lie groups to Lie algebras I hope that functoriality may still be achieved with some extra conditions on both sides of the mapping.

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  • $\begingroup$ the non-uniqueness suggests considering equivalence classes or have you already? see my later comment further down in the list $\endgroup$
    – Jim Stasheff
    Commented Jul 4, 2019 at 14:54

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Hi Igor,

there is a quite elementary way to see that star products restrict to open subsets: it's essentially part of the definition of a star product. Here, I will focus on the case of smooth (symplectic/Poisson) manifolds, where you do not have to rely too much on sheaf-theoretic notions. Now, $\star$ being a star product on $M$ means that it consists in each order of $\hbar$ of bidifferential operators if you consider a differential star product, or of local operators if you consider a local star product. Local operators are, by a celebrated theorem of Preetre locally differential, i.e. on a sufficiently small open subset the restrict to differential operators. Globally, the only problem may be that their order is infinite (take the real line and a bump function with supp in the unit interval. Now translate the bump function to a bump function $\chi_n$ having supp in $[n,n+1]$ Then taking the sum of all $\chi_n \frac{\partial^n}{\partial x^n}$ is a local but not a differential operator). In any case, bidifferential/local operators restrict to open subsets.

THis induced for every open $U \subseteq M$ a new product $\star_U$ for $C^\infty(U)[[\hbar]]$. The point is now that associativity can be check locally, as in each order of $\hbar$ it is an equation between (multi-)differential operators which localize! So you indeed get a star product.

Unfortunately, with the advent of Kontsevich's formality these things and techniques were mostly forgotten. It was kind of standard arguments and it's also present in all the earlier constructions of/with star products in the 80's... It might be that it is not spelled out here explicitly in the literature, but you just have to look at the early papers of Bayen etc as well as Gutt, Cahen, deWilde, Lecomte and so on, and scan for the words "local operator" or Peetre Theorem ;)

Now for the second question about functoriality. This is of course much more subtle. The naive answer is that there is no thing like a quantization functor (with appropriate continuity properties) as one has the no-go theorem of Gronewold and van Hove. I'm sure you know this. So one has to refine things a bit: the crucial question seems to be what the domain of this functor should be: symplectic manifolds per se is not a good choice. A better choice will be symplectic manifolds with a symplectic connection (this is a huge choice to make, as there are zillions of symplectic connections...)

In this case, one indeed gets a functorial quantization by means of the Fedosov construction of a star product (say for trivial characteristic class) This you can find at many places in the literatur (if you're crazy enough to read german, you can take a look at my book ;) As Theo say, one can modify the construction using a series of closed two-forms, so it might be better to chose this as a domain of you functor...

Simillar things hold in the Poisson case as well: you have to choose a formality for $\mathbb{R}^n$ once and for all, say the Kontsevich one, subject to certain invariance properties ($\mathrm{GL}(n)$-invariant should do the job). Then you can globalize this formality by Dolgushev's construction. This gives a functorial construction of a formality for the price of choosing a connection on each manifold first (this is needed in Dolgushev). The again, one get's functorial star products... This is sort of implicit in a paper by Kontsevich. If I remember correctly, in an appendix... BUt more details on it are in the papers by Dolguushev.

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  • $\begingroup$ H, Stefan. Thanks a lot for your answer! I suspected that the argument via locality of differential operators should go through, but it's nice to know some references where this is at least mentioned. Now, as I understand it, the restriction to an open subset induces a homomorphism between the deformed algebras, which implies the existence of a two-sided ideal in the algebra of the larger region. For definiteness, consider $M=\mathbb{R}^2$ and the unit disk $N\subset M$. Let $I_N\subset (\mathbb{R}^2,\star)$ be the two-sided ideal corresponding to the restriction to $N$. ... $\endgroup$
    – Igor Khavkine
    Commented Sep 19, 2011 at 8:54
  • $\begingroup$ ... Doesn't then the existence of $I_N$ interfere with the uniqueness result of the Stone-von Neumann theorem? I realize that this is at most a moral objection, since the S-vN theorem applies to the Weyl algebra associated to $(\mathbb{R}^2,\star)$ and is outside the realm of formal power series in $\hbar$. I guess I'm not seeing clearly what happens to $I_N$ through the construction of the Weyl algebra and making the deformation strict. What can then also be said about representations of $(N,\star)$ (or the Weyl algebra thereof)? $\endgroup$
    – Igor Khavkine
    Commented Sep 19, 2011 at 9:03
  • $\begingroup$ I guess this is really a matter of convergence and analysis. If you look at the SvN-Theorem it heavily uses the whole $C^*$-algebraic machinery. In fact, for formal star products there is no uniqueness of a representation at all. You can construct easily very very inequivalent representations (I have a long review article on that...) However, from a more physical point of view I would consider that as an artefact of the purely algebraic nature of formal DQ. So as soon as you add some convergence and perform the necessary topological closures, the inequivalences should disappear... $\endgroup$
    – Stefan Waldmann
    Commented Sep 22, 2011 at 12:49
  • $\begingroup$ ... But this is of course only a ideological answer as in general there is no theorem stating what I claim. In fact, it is not at all clear what you mean by "convergence" and "topological closure": one knows a few (very few) examples where things can be done but unfortunately there is no general theory of what people call "strict" DQ, no classification etc... So in that direction there is still a lot of work to be done ;) $\endgroup$
    – Stefan Waldmann
    Commented Sep 22, 2011 at 12:51
  • $\begingroup$ OK, I think I see now that the problem really lies in the difference between formal and strict deformations. Since this is enough information for me to now know why my intuition was making me uneasy, I'll mark this as answered. $\endgroup$
    – Igor Khavkine
    Commented Sep 22, 2011 at 20:55
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Many definitions I've seen of "star quantization of a Poisson manifold" include the request that each coefficient in $\hbar$ of $a\star b$ be a differential operator in $a,b$. Such a star quantization will will certainly restrict to any open set. It's important to emphasize that you can only expect such restrictions in the perturbative regime. In nonperturbative quantization, you should expect that different locations on the manifold simply have nontrivial "interference", and so you would not expect a good theory of restriction to opens.

If memory serves, in the symplectic case the space of star quantizations of a manifold is a torsor for the space of closed de Rham 2-forms with coefficients in $\mathbb R[\\![ \hbar ] \\!]$. (Certainly there's a natural nonempty space of isomorphisms between the space of quantizations and the space of closed 2-forms; I don't remember if these isomorphisms respect enough of the abelian group structure to really warrant the word "torsor".) So at least in the symplectic case it shouldn't be too hard to write down the "stack of quantizations", by relying on results by Kontsevich and later. (Actually, existence of star quantization is much older than Kontsevich in the symplectic case, but Kontsevich clarified the structure of the space of all quantizations.) The heart of the story is some of the formality theorems, and about here one should also mention the name Tamarkin.

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The deformation quantizations of a symplectic manifold (at least in the complex analytic case) form an algebroid stack. This is a certain kind of gerbe with a sheaf of algebras over it. One place where this was proved is in this paper, by Polesello-Schapira, although the idea goes back to Kontsevich, as far as I know. I find these papers very hard to follow, and if someone has easier references, I would also like to know.

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