Index theory for basic Dirac operators on Riemannian foliations

TL;DR

This paper derives an index formula for basic Dirac operators on Riemannian foliations, involving integrals over foliation strata and eta invariants, and includes a Gauss-Bonnet type result for the basic Euler characteristic.

Contribution

It provides the first explicit index formula for basic Dirac operators on Riemannian foliations, solving a longstanding open problem.

Findings

Derived a formula involving integrals over foliation strata and eta invariants.

Obtained a Gauss-Bonnet formula for the basic Euler characteristic.

Extended the index theory to twisted basic Dirac operators.

Abstract

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.