Taking topological field theory at phase value
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In this thesis, we use methods of topological field theory to model and study topological phases of matter. This includes computing TFTs that capture low-energy information for the GDS model and for the Majorana chain with time-reversal symmetry. We then investigate phases of matter with spatial symmetries that mix with the internal symmetry type; we provide a mathematical model for these phases and prove a “fermionic crystalline equivalence principle” theorem as predicted in the physics literature. Some of our computations lead to a bonus theorem on the classification of some unorientable 4-manifolds up to stable diffeomorphism.