The families Clifford index and differential KO-theory

Extending ideas of Atiyah–Bott–Shapiro and Quillen, we construct a model for differential $\mathrm{K}\mathrm{O}$-theory whose cocycles are families of Clifford modules with superconnection. The model is built to accommodate an analytic pushforward for bundles of spin manifolds, affording a differential refinement of Atiyah and Singer’s families index.