C-R-T Fractionalization, Fermions, and Mod 8 Periodicity

Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion parity (-1)^ F are basic discrete spacetime and internal symmetries of the Dirac fermions. In this article, we determine the group, called the C-R-T fractionalization, which is a group extension of ℤ_2^ C×ℤ_2^ R×ℤ_2^ T by the fermion parity ℤ_2^ F, and its extension class in all spacetime dimensions d, for a single-particle fermion theory. For Dirac fermions, with the canonical CRT symmetry ℤ_2^ CRT, the C-R-T fractionalization has two possibilities that only depend on spacetime dimensions d modulo 8, which are order-16 nonabelian groups, including the famous Pauli group. For Majorana fermions, we determine the R-T fractionalization in all spacetime dimensions d=0,1,2,3,48, which is an order-8 abelian or nonabelian group. For Weyl fermions, we determine the C or T fractionalization in all even spacetime dimensions d, which is an order-4 abelian group. For Majorana-Weyl fermions, we only have an order-2 ℤ_2^ F group. We discuss how the Dirac and Majorana mass terms break the symmetries C, R, or T. We study the domain wall dimensional reduction of the fermions and their C-R-T fractionalization: from d-dim Dirac to (d-1)-dim Dirac or Weyl; and from d-dim Majorana to (d-1)-dim Majorana or Majorana-Weyl.