An 8-fold way for CRT

Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the ℤ2×ℤ2×ℤ2 for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity, thereby forming a group extension of the ℤ2 direct product. Conventionally, a Majorana fermion is defined by one Dirac fermion with trivial charge conjugation. However, when the spacetime dimension d+1=5,6,7mod8, the real dimension of Majorana fermion (dimℝχℓ(d,0)) aligns with the real dimension of Dirac fermion (dimℝψℓ(d)), rather than being half, which necessitates the introduction of a symplectic Majorana fermion, defined by two Dirac fermions with trivial charge conjugation. To include these two types of Majorana fermions, we embed the theory in nℝ and define the Majorana fermion field as a representation of the real Clifford algebra with 8-fold periodicity. Within the Hamiltonian formalism, we identify the 8-fold CRT-internal symmetry groups across general dimensions. Similarly, Dirac fermion field is defined as a representation of the complex Clifford algebra with 2-fold periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an 8-fold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act non-trivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.