Diracs quatization condition is usually treated in quantum field theory classes.

The bundle talked about is treated in manifold classes, sometimes differential geometry (although that is strictly more than you need), but often bundles are just covered in QFT/string theory classes on a physics level of rigor. Moreover, this seems to be about string theory, which you would need to take.

So you would first need a lot of calculus, linear algebra, complex analysis, analysis in many dimensions, advanced QM, E&M, classical field theory, advanced thermal physics, group theory, and maybe even some topology.

In short, you need the basic education of any theoretical physicist