Instanton-Like Solutions of Euclidean Supersymmetric Yang-Mills Theory

Abstract

In recent years considerable interest has been given to the supersymmetric extension of Yang-Mills theories. l ) The N=4 supersymmetric theory in d=4 dimensions is the maximally extended supersymmetric Yang-Mills theory and is closely related to the N=l supersymmetric Yang-Mills theory in d=10 dimensions.)-5) The discovery of pseudoparticle (instanton) solutions to the Euclidean SU(2) Yang-Mills theory6) has suggested the possibility that analogous solutions might occur in Euclidean gravity7) as well as in the supersymmetric extensions of Yang-Mills.) Since the YangMills instantons' posses self-dual field strength, one likely possibility is that gravitational instantons are characterized by self-dual curvature. In papers published previouslyll)-13) we presented various self-dual solutions of Euclidean gravity. Besides being new exact solutions to Einstein's field equations, the solutions given are double self-dual solutions of 0(4) Yang-Mills equations .. In this paper we consider the bosonic sector of the Euclidean N =4 supersymmetric Yang-Mills theory with global 0(4) symmetry. We consider Euclidean superspace with coordinates (xl', 8 s, (flit); ,u=O, 1,2,3; s, t = 1, 2; 3, 4; a, t = 1, 2, where xl' are the Cartesian coordinates of even space-time variables and e, (f are the odd spinor coordinates. In that case e and (f are not related by complex conjugation as in Lorentz superspace. In Euclidean superspace one may, however, use hermiticity under the unitary involutary operator of Osterwalder and Schrader.14),15) The superfields (j)(x, e, 1J) are then expressed only in terms of e and (f as one has for the Lorentz superfields. We consider the Yang-Mills superpotential (superconnection) A with values in the Lie algebra