Higher localized analytic indices and strict deformation quantization

TL;DR

This paper develops a framework for localizing higher analytic indices of Lie groupoids using deformation quantization, providing explicit formulas and unifying previous index theories.

Contribution

It introduces a method to localize higher analytic indices via strict deformation quantization, extending previous results to étale groupoids and unifying various index formulas.

Findings

Every bounded cyclic cocycle can be localized.

For étale groupoids, all cyclic cocycles can be localized.

Provides explicit formulas for higher localized indices using asymptotic limits.

Abstract

This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let $\gr$ be a Lie groupoid with Lie algebroid $A\gr$. Let $\tau$ be a (periodic) cyclic cocycle over the convolution algebra $\cg$. We say that $\tau$ can be localized if there is a correspondence K^0(A^*\gr)\stackrel{Ind_{\tau}}{\longrightarrow}\mathbb{C} satisfying $Ind_{\tau}(a)=< ind D_a,\tau>$ (Connes pairing). In this case, we call $Ind_{\tau}$ the higher localized index associated to $\tau$. In {Ca4} we use the algebra of functions over the tangent groupoid introduced in {Ca2}, which is in fact a strict deformation quantization of the Schwartz algebra $\sw(A\gr)$, to prove the following results: \item Every bounded continuous cyclic cocycle can be localized. \item If $\gr$ is {\'e}tale, every cyclic cocycle can be localized. We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting.