2025/11/25 by Su, Tan
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows
paper · doi:10.48550/arxiv.2511.19904
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.