Localization over complex-analytic groupoids and conformal renormalization

TL;DR

This paper develops a higher index theorem for complex-analytic groupoids using conformal renormalization, linking local anomalies to non-commutative geometry and cyclic cocycles.

Contribution

It introduces a conformal renormalization approach for index theory on complex-analytic groupoids, utilizing local anomaly formulas and cyclic cocycles.

Findings

Index formula localized at automorphisms

Use of cyclic cocycles including a non-commutative Todd class

Connection between anomalies and index theory in non-metric geometry

Abstract

We present a higher index theorem for a certain class of etale one-dimensional complex-analytic groupoids. The novelty is the use of the local anomaly formula established in a previous paper, which represents the bivariant Chern character of a quasihomomorphism as the chiral anomaly associated to a renormalized non-commutative chiral field theory. In the present situation the geometry is non-metric and the corresponding field theory can be renormalized in a purely conformal way, by exploiting the complex-analytic structure of the groupoid only. The index formula is automatically localized at the automorphism subset of the groupoid and involves a cap-product with the sum of two different cyclic cocycles over the groupoid algebra. The first cocycle is a trace involving a generalization of the Lefschetz numbers to higher-order fixed points. The second cocycle is a non-commutative Todd class, constructed from the modular automorphism group of the algebra.