to and the imaginary part to . This doesn't even exhaust all the possibilities. We see that there is more than one N = 2 supersymmetry; likewise, the SUSYs for N > 2 are also not unique (in fact, it only gets worse). N = 3 It is theoretically allowed, but the multiplet structure becomes automatically the same with that of an N=4 supersymmetric theory. So it is less often discussed compared to N=1,2,4 version. N = 4 This is the maximal number of supercharges in a theory without gravity. SUSY in various dimensions In 0 + 1, 2 + 1, 3 + 1, 4 + 1, 6 + 1, 7 + 1, 8 + 1, 10 + 1 dimensions, etc., a SUSY algebra is classified by a positive integer N. In 1 + 1, 5 + 1, 9 + 1 dimensions, etc., a SUSY algebra is classified by two nonnegative integers (M, N), at least one of which is nonzero. M represents the number of left-handed SUSYs and N represents the number of right-handed SUSYs. The reason of this has to do with the reality conditions of the spinors. Hereafter d = 9 means d = 8 + 1 in Minkowski signature, etc. The structure of supersymmetry algebra is mainly determined by the number of the fermionic generators, that is the number N times the real dimension of the spinor in d dimensions. It is because one can obtain a supersymmetry algebra of lower dimension easily from that of higher dimensionality by the use of dimensional reduction. d = 11 The only example is the N = 1 supersymmetry with 32 supercharges. d = 10 From d = 11, N = 1 SUSY, one obtains N = (1, 1) nonchiral SUSY algebra, which is also called the type IIA supersymmetry. There is also N = (2, 0) SUSY algebra, which is called the type IIB supersymmetry. Both of them have 32 supercharges. N = (1, 0) SUSY algebra with 16 supercharges is the minimal susy algebra in 10 dimensions. It is also called the type I supersymmetry. Type IIA / IIB / I superstring theory has the SUSY algebra of the corresponding name. The supersymmetry algebra for the heterotic superstrings is that of type I. Remarks Notes References Supersymmetry Lie algebras