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

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.