In Kontsevich's Deformation quantization of Poisson manifolds, he gives an explicit formula for the star product: $$ f \star g = fg + \sum_{n=1}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma} (f, g) \tag{$\ast$} $$ where $B_\Gamma (f, g)$ is the bilinear operator associated to a graph $\Gamma$ and $w_\Gamma$ is a "weight", constructed as the integral of some configuration space.

I understand that the generalization of Kontsevich formality to the algebraic setting (smooth algebraic variety, possibly non-affine) is more involved and I have the following questions:

  1. Is the formula for the weights in the (complex) algebraic context exactly the same as in the real and smooth case?

  2. Are the weights computed "globally" or chart-by-chart, so that one has to check compatibility/gluing in an additional step?

  1. yes, the formula for the weights is the same in whatever setting (differentiable, holomorphic and algebro-geometric).

  2. weights are involved in a local formula for a start-product. Indeed, $B_\Gamma$ doesn't even make sense globally on a manifold (being differentiable, holomorphic, etc...). Hence, even in the differentiable setting, Kontsevich's local formula must be globalized. There are standard techniques to do this, but the resulting star-product is not quite explicit.

Let me add that in the holomorphic (or algebro-geometric) setting there is an additional issue: a global star-product may not exist. Indeed, it is not always possible to glue the local formula in order to get a sheaf of algebras deforming the structure sheaf of the complex manifold/algebraic variety. As it is explained in this other paper of Kontsevich, one can only hope to get an algebroid stack (this is a linearized analog of a gerbe). There has been a lot of work on these gadgets (by Yekutieli, Kashiwara-Schapira, and many others).

  • 1
    $\begingroup$ Thank you very much for your answer. Do you know of any papers that give explicit star products on non-local varieties? (I guess in the symplectic case such a quantization could be obtained for $T^* X$ as some form of $\mathscr D_X$, but I also haven't seen the corresponding star product being given explicitly.) $\endgroup$ – Earthliŋ Jan 22 '18 at 8:24
  • $\begingroup$ Consider a fibre bundle $E\to X$ together with a Poisson structure on its total space that is linear in the fibers. Then $E$ is dual to a Lie algebroid $L$. And the universal enveloping algebra of $L$ gives a deformation quantization of the fiberwise linear Poisson structure on $E$. $\endgroup$ – DamienC Feb 1 '18 at 11:59
  • $\begingroup$ Would you have a reference for this? My understanding is that as a deformation quantization with parameter $\hbar$, say, one wants to consider a "homogenized" universal enveloping algebra (homogenizing the relations using the parameter $\hbar$), giving the usual enveloping algebra for $\hbar = 1$. But I haven't seen an explicit description of the bidifferential operators appearing in the star product that describe this quantization. $\endgroup$ – Earthliŋ Feb 1 '18 at 12:48
  • 1
    $\begingroup$ I don't have a reference in mind. Of course one has to introduce $\hbar$ in the defining relations for the universal envelopping algebra: $e_1e_2-e_2e_1=\hbar[e_1,e_2]$, $ef-fe=\hbar\rho(e)(f)$. The fact that the coefficients of the deformation are bidifferential operators is a property that can be checked locally, and that is a consequence of the PBW theorem for Lie algebroids (see e.g. the paper of Nistor-Weinstein-Xu: arxiv.org/pdf/funct-an/9702004.pdf). $\endgroup$ – DamienC Feb 14 '18 at 18:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.