Flux quantization and the M-theoretic characters

In a previous work [H. Sati, M-theory and characteristic classes, JHEP 0508 (2005) 020, hep-th/0501245] we introduced characters and classes built out of the M-theory four-form and the Pontrjagin classes, which we used to express the Chern-Simons and the one-loop terms in a way that makes the topological structures behind them more transparent. In this paper we further investigate such classes and the corresponding candidate generalized cohomology theories. In particular, we study the flux quantization conditions that arise in this context.