HIGH SYMMETRY SOLUTIONS OF THE ANTI-SELF-DUAL YANG-MILLS EQUATIONS

Beginning with the anti-self-dual Yang-Mills (ASDYM) equations for an arbitrary Lie algebra on Minkowski space, this paper specializes to the case in which the vector potentials are independent of all the space-time coordinates, i.e., are space-time constants. The resulting equations are three algebraic equations on the algebra. These equations are then simplified by using a null basis. Two of the equations can be immediately solved while the third remains, in general, quite difficult to deal with. Two general cases are considered: finite-dimensional Lie groups and the infinite-dimensional diffeomorphism groups on finite-dimensional manifolds. In a few of the special cases, e.g., SL(2,C) and the Virasoro algebra, the solutions can easily be found. The study of the the diffeomorphism groups leads unexpectedly to the Monge-Ampère equation. In particular, the four-dimensional volume preserving diffeomorphism group is identical with the vacuum anti-self-dual Einstein equations. In conclusion, the question of the associated Lax pair equations and its relation to the Riemann-Hilbert splitting problem on S2 is examined. © 1990 American Institute of Physics.