Twisted differential KO-theory

We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose description involves appropriate local systems. Along the way, we provide a complete and explicit identification of the differentials at the $E_2$ and $E_3$ pages in the topological case, which has been missing in the literature and which is needed for the general case. The corresponding differentials in the refined theory reveal an intricate interplay between topological and geometric data, the former involving the flat part and the latter requiring the construction of the twisted differential Pontrjagin character. We illustrate with examples and applications from geometry, topology and physics. For instance, quantization conditions show how to lift differential $4k$-forms to twisted differential KO-theory leading to integrality results, while considerations of anomalies in type I string theory allow for characterization of twisted differential Spin structures.