Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8

TL;DR

This paper develops a differential KO-theory framework in degrees divisible by 8 using eta-invariants and index theorems, extending previous work in differential K-theory to real vector bundles and spin fiber bundles.

Contribution

It introduces a differential KO-character using eta-invariants and establishes two family index theorems in differential KO-theory for spin fiber bundles.

Findings

Complete determination of differential KO-theory in degrees 0 mod 8 via eta-invariants.

Development of two compatible family index theorems in differential KO-theory.

New interpretation of the Bismut--Cheeger adiabatic limit theorem.

Abstract

Sullivan--Simons developed a Cheeger--Simons differential character analogue for degree (0 mod 2) differential K-theory, giving a complete set of numerical invariants that determine a complex vector bundle with unitary connection on a base manifold X, up to Chern--Simons equivalence of the connection. In this paper we develop a degree (0 mod 8) differential KO-analogue. Namely, given a real vector bundle with orthogonal connection, we construct R/Z -valued eta-invariants in the context of Atiyah--Patodi--Singer and Z2 Atiyah--Singer index invariants that completely determine differential KO-theory in degree (0 mod 8); we call this the differential KO-character. In the second part, for a Riemannian submersion X to B with closed 8k-dimensional spin fibers, we develop two family index theorems in differential KO -theory: one in the differential KO -character model and one in the structured-bundle model. The fact that these two pushforwards agree follows from the Bismut--Cheeger adiabatic limit theorem, providing a new interpretation of that result.