A symbol calculus for foliations

TL;DR

This paper extends the Getzler rescaling theorem to the transverse geometry of foliations, developing a calculus for transversely spin foliations to better understand their index theory.

Contribution

It introduces a Getzler rescaling and Block-Fox calculus for all transversely spin foliations, enabling analysis of transversely elliptic operators.

Findings

Composition of AΨDOs remains an AΨDO with a specified leading symbol.

Provides a calculus applicable to operators of mixed degrees in leaf and transverse directions.

Enhances understanding of index theory for transversely elliptic operators on foliations.

Abstract

The classical Getzler rescaling theorem is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, as well as a Block-Fox calculus of asymptotic operators, is constructed for all transversely spin foliations. This calculus applies to operators of degree $m$ globally times degree $\ell$ in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations. The main result is that the composition of A$\Psi$DOs is again an A$\Psi$DO, and includes a formula for the leading symbol.