Associative half-densities on symplectic groupoids and quantization

TL;DR

This paper develops a theory of associative half-densities on symplectic groupoids, linking them to semiclassical quantization and extending known results like the Duflo isomorphism.

Contribution

It introduces and classifies associative half-densities on symplectic groupoids, connecting them to semiclassical star products and quantization formulas.

Findings

Existence and classification of associative half-densities established.

Application to semiclassical factors in Kontsevich's quantization.

Recovery of Duflo isomorphism factors in the linear Poisson case.

Abstract

In this paper, we study half-densities enhancing the multiplication map on a symplectic groupoid and which satisfy a suitable associativity condition. This is structurally motivated by the expected complete semiclassical-analytic approximation to a star product for the underlying Poisson manifold. We show the existence and classification of such associative half-densities, and further apply this theory to the understanding of semiclassical factors in Kontsevich's quantization formula. In the particular case of a linear Poisson structure, we recover the factors appearing in the Duflo isomorphism and its Kashiwara-Vergne extensions as a canonical associative enhancement.