High symmetry solutions of the anti-self-dual Yang–Mills equations

Abstract

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.