I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is known that, in this case, the first Chern number describes the topological invariant for the entire Brillouin zone. However, what happens if we take a closed adiabatic loop in the Brillouin torus associated with Haldane's model? Then we have a manifold with a boundary.
I might be wrong, but to my understanding, the linked reference suggests, in equations 8.2 and 8.5 that the Atiyah-Singer index theorem implies that the second Chern class will be involved in this case and that we will still have a quantized topological invariant.
However, I am having trouble wrapping my head around this because Haldane's model is a 2D system, and other resources (like the top-most paragraph in page 1630003-19 of this paper, for example) imply that it is not useful to look at the second Chern number in 2D. What then, does this topological invariant mean? Won't it be useless if it is not applicable?
Clearly, there is a gap in my understanding, so I would appreciate it if someone could help me see what I missed. Thanks!
Reference 1: Gravitation, Gauge Theories and Differential Geometry, by Eguchi, Gilkey & Hanson
Reference 2: Notes on topological insulators, by R. Kaufmann & D. Li