Extended Supersymmetry in Four-Dimensional Euclidean Space

Since the generators of the two SU(2) groups which comprise SO(4) are not Hermitian conjugates of each other, the simplest supersymmetry algebra in four-dimensional Euclidean space more closely resembles the N = 2 than the N = 1 supersymmetry algebra in four-dimensional Minkowski space. An extended supersymmetry algebra in four-dimensional Euclidean space is considered in this paper; its structure resembles that of N = 4 supersymmetry in four-dimensional Minkowski space. The relationship of this algebra to the algebra found by dimensionally reducing the N = 1 supersymmetry algebra in ten-dimensional Euclidean space to four-dimensional Euclidean space is examined. The dimensional reduction of N = 1 super Yang-Mills theory in ten-dimensional Minkowski space to four-dimensional Euclidean space is also considered.