Linear field equations on self-dual spaces

Abstract

In a Riemannian context, a description is given of the Penrose correspondence between solutions of the anti-self-dual zero rest-mass field equations in a self-dual Yang-Mills background on a self-dual space X, and the sheaf cohomology groups H 1 ( Z, OF ( n )), for n ≤ -2of its twistor space Z . The case n = - 2 is fundamental for the construction of instantons on Euclidean space. It is further shown how H 1 ( Z, OF (-1)) corresponds to solutions of the self-dual Dirac equation, and an interpretation for H 1 ( Z, OF ( n )), for n ≥ 0, is given in terms of the cohomology of an elliptic complex on X .