Deformation quantization with traces

TL;DR

This paper proves that for divergence-free Poisson structures, the Kontsevich star-product is cyclic, and extends these results globally to Poisson manifolds, also addressing the Connes-Flato-Sternheimer conjecture.

Contribution

It establishes the cyclicity of the Kontsevich star-product for divergence-free Poisson structures and generalizes this to arbitrary Poisson manifolds, also proving a related conjecture.

Findings

Kontsevich star-product is cyclic for divergence-free Poisson bivectors

Globalization of cyclicity for arbitrary Poisson manifolds

Proof of the Connes-Flato-Sternheimer conjecture in the Poisson case

Abstract

In the present paper we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a divergence-free Poisson bivector field on R^d, the Kontsevich star-product with the harmonic angle function is cyclic. We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and prove a generalization of the Connes-Flato-Sternheimer conjecture on closed star-products in the Poisson case.