Trace Functionals of the Kontsevich Quantization

TL;DR

This paper extends the concept of trace in Kontsevich quantization, demonstrating that for certain Poisson manifolds, the trace corresponds to integration over a unimodular volume form, linking quantization and geometric measures.

Contribution

It generalizes the notion of trace in Kontsevich quantization to a broader class of Poisson manifolds, specifically those related to quotients of symplectic manifolds by nilpotent Lie group actions.

Findings

Trace in Kontsevich quantization equals integration over unimodular volume forms for specific Poisson manifolds.

Provides a geometric interpretation of the trace in the context of Hamiltonian quotients.

Extends previous notions of trace to include a wider class of Poisson structures.

Abstract

We generalize the notion of trace to the Kontsevich quantization algebra and show that for all Poisson manifolds representable by quotients of a symplectic manifold by a Hamiltonian action of a nilpotent Lie group, the trace is given by integration with respect to a unimodular volume form.