A K-Theoretic Framework for Neutral Fermionic Topological Phases

Freed and Moore's "Twisted Equivariant Matter" establishes a K-theory-based classification for free fermion symmetry-protected topological phases, building on the work of Kitaev. I present a generalization of their approach which does not assume the protecting symmetry group contains a $U(1)$. I will explain how the tenfold way arises using Wall's classification of $\mathbb{Z}_2$-graded division algebras. If time permits, I will explain how it relates to the tenfold way one can find in Freed-Moore, originally due to Ryu-Schnyder-Furusaki-Ludwig.