Actions for spinor fields in arbitrary dimensions

We are going to give a systematic presentation of spinors in various spacetime dimensions which is a prerequisite for understanding the supersymmetry. (For general reviews see [1,2,3].) To set up the conventions, we will consider in what follows the Dirac matrices that satisfy the following Clifford algebra {γ, γ} = 2η1 , η = diag(+ · · ·+,− · · ·−) , (1) the sign +, − appear t, s times which are the numbers of timelike, spacelike directions, respectively. The Hermitian, complex and transpose conjugation operations are defined as usual [4,5] and are given by the relations γ = −(−1)AγA , γ = ηBγB , γ = −η(−1)CγC , (2) where η = ±1 corresponds to independent choices of charge conjugation matrix. A, B and C are unitary matrices and the following relations hold among them A = γ...γ , B T = εB , C ≡ B T A . (3) Here, 1, . . . , t are all timelike directions and