I am currently trying to go through some literature on the classification of symmetry protected topological phases. Primarily, I am interested in the classical of topological phases using mathematical tools: group cohomology, cobordism, higher category theory etc.
One shortcoming in my background is that I am unable to understand the physics-motivated discussions of gauge theory in the context of condensed matter physics. What is a background gauge field? What are emergent gauge fields? What is symmetry-breaking? Etc.
I am looking for mathematically precise references that I can use to learn the relevant gauge theory (both with discrete and Lie groups, I guess) so I can better understand the literature on classification of topological phases.
Any help would be appreciated!