On the algebraic index for riemannian \'etale groupoids

TL;DR

This paper develops an algebraic index theory for Riemannian étale groupoids using explicit quasi-isomorphisms to analyze cyclic cohomology in deformation quantization, addressing the index problem in specific cases.

Contribution

It introduces a new algebraic index framework for Riemannian étale groupoids and provides solutions in particular geometric settings.

Findings

Constructed explicit quasi-isomorphism for cyclic cohomology analysis

Formulated a general algebraic index problem for Riemannian étale groupoids

Solved the index problem for proper groupoids and constant Dirac structures on a 3-torus

Abstract

In this paper we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a riemannian \'etale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for riemannian \'etale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dim torus.