Deformation quantization with traces. (English) Zbl 0983.53065

Felder, Giovanni; Shoikhet, Boris

Lett. Math. Phys. 53, No. 1, 75-86 (2000).

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.

MSC:

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

Keywords:

cyclic formality conjecture; constant volume form; Poisson bivector field; Kontsevich star product; harmonic angle function; Poisson manifolds; closed star products