Deformation quantization with traces. (English) Zbl 0983.53065

Summary: In this letter we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a constant volume form \(\Omega\) and a Poisson bivector field \(\pi\) on \(\mathbb{R}^d\) such that \(\text{div}_\Omega \pi=0\), the Kontsevich star product with the harmonic angle function is cyclic, i.e., \(\int_{\mathbb{R}^d}(f*g)\cdot h\cdot \Omega= \int_{\mathbb{R}^d} (g*h)\cdot f\cdot \Omega\) for any three functions \(f,g,h\) on \(\mathbb{R}^d\) (for which the integrals make sense). We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and an arbitrary volume form, and prove a generalization of the Connes-Flato-Sternheimer conjecture on closed star products in the Poisson case.


53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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