Quantization of linear Poisson structures and degrees of maps.

*(English)*Zbl 1056.53060M. Kontsevich [Deformation quantization of Poisson manifolds. I. Lett. Math. Phys. 66, No. 3, 157–216 (2003; Zbl 1058.53065), see also q-alg/9709040 (1997)] provided a deformation quantization of the algebra of functions on a Poisson manifold. An important problem related to this construction for \(\mathbb{R}^{d}\) is to interpret the weights of graphs geometrically, and to show a way to compute them. Natural questions are whether 1-forms on \(S^{1}\) lead to an associative star product, and if yes, how do these products depend on the choice of a form.

The paper answers both questions in the linear case. It is shown that any 1-form on \(S^{1}\) gives an associative star product. This is done by an interpretation of the associativity as a vanishing of some cocycle in the relative top cohomology of a grand configuration space \(C_{n,3}\), obtained by gluing together the configuration spaces corresponding to graphs along their boundary strata. By choosing singular forms concentrated in one point on a torus, one gets star products with rational coefficients which can be computed by a simple combinatorial count. The problem of comparison of these star products is solved in a similar fashion, by identifying terms with some fixed top cohomology class of a another grand configuration space \(C_{n,2}\).

The paper answers both questions in the linear case. It is shown that any 1-form on \(S^{1}\) gives an associative star product. This is done by an interpretation of the associativity as a vanishing of some cocycle in the relative top cohomology of a grand configuration space \(C_{n,3}\), obtained by gluing together the configuration spaces corresponding to graphs along their boundary strata. By choosing singular forms concentrated in one point on a torus, one gets star products with rational coefficients which can be computed by a simple combinatorial count. The problem of comparison of these star products is solved in a similar fashion, by identifying terms with some fixed top cohomology class of a another grand configuration space \(C_{n,2}\).

Reviewer: Gheorghe Pitiş (Braşov)

##### MSC:

53D55 | Deformation quantization, star products |

55R80 | Discriminantal varieties and configuration spaces in algebraic topology |

57R35 | Differentiable mappings in differential topology |