In some recent papers, such as Zohar PhysRevB.97.054418, Zohar arXiv:1705.04786, Metlitski PhysRevB.98.085140, the authors state that the anomaly inflow term/ topological action can be expressed in cocycles, and these terms have the form like $\int A\wedge w_{2}$, where $A$ is the gauge field associating with the global symmetry involved in mixed 't Hooft anomaly, $w_{2}$ is the second Stiefel-Whitney class.
I don't know why 't Hooft anomaly can be described by these characteristic class, are there references for this method? And I can't find this method in David Tong's gauge theory lecture notes, Nakahara's book, or some quantum field theory book.
