Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Kontsevich $\star$-product associativity mechanism   The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to
construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\
\}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a
given finite-dimensional affine Poisson manifold: here $\times$ is the usual
multiplication, $\{\ ,\ \}_{P}\neq0$ is the Poisson bracket, and $\hbar$ is the
deformation parameter. The product $\star$ is assembled at all powers
$\hbar^{k\geq0}$ via summation over a certain set of weighted graphs with $k+2$
vertices; for each $k>0$, every such graph connects the two co-multiples of
$\star$ using $k$ copies of $\{\ ,\ \}_{P}$. Cattaneo and Felder [
arXiv:math/9902090 [math.QA] ] interpreted these topological portraits as the
genuine Feynman diagrams in the Ikeda-Izawa model [arXiv:hep-th/9312059] for
quantum gravity.
  By expanding the star-product up to $\bar{o}(\hbar^3)$, i.e., with respect to
graphs with at most five vertices but possibly containing loops, we illustrate
the mechanism Assoc = Operator(Poisson) that converts the Jacobi identity for
the bracket $\{\ ,\ \}_{P}$ into the associativity of $\star$.
  Key words: Deformation quantization, associative algebra, Poisson bracket,
graph complex, star-product
  PACS: 02.40.Sf, 02.10.Ox, 02.40.Gh, also 04.60.-m

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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc