Transversal Dirac operators on distributions, foliations, and G-manifolds: Lecture notes

TL;DR

This survey explores the geometric and analytic aspects of transverse Dirac operators on distributions, foliations, and G-manifolds, highlighting their spectral invariance, index formulas, and self-adjointness properties.

Contribution

It introduces a definition of transverse Dirac operators for distributions, proves spectral invariance, and provides Atiyah-Singer type index formulas for transversally elliptic operators.

Findings

Transverse Dirac operators are essentially self-adjoint.

Spectral invariance of basic Dirac operators on Riemannian foliations.

Atiyah-Singer type index formulas for transversally elliptic operators.

Abstract

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spectrum of a basic Dirac operator on a Riemannian foliation. The Bruening-Kamber-R theorems give Atiyah-Singer type formulas for the equivariant index of transversally elliptic operators on G-manifolds and the index of basic Dirac operators on Riemannian foliations. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.