Paul Tod, University of Oxford
The twistor construction of black holes and gravitational instantons relies on two results: the first, due to Richard Ward, that there is a one-to-one correspondence between solutions of the anti-self-dual Yang-Mills equations (ASDYM) on (part of) flat space-time and holomorphic vector bundles on (part of) complex projective 3-space (CP3) which are ‘trivial on lines’ i.e. on CP1’s in CP3; the second, due to Lou Witten and picked up by Richard Ward, that the Einstein vacuum equations for a stationary, axisymmetric 4-dimensional black hole essentially reduce to a particular case of the static, axisymmetric ASDYM equations on flat space-time. Putting these together, the possibility immediately arises of constructing vacuum black hole solutions in terms of holomorphic vector bundles on CP3 reduced by the two symmetries, a one-dimensional complex manifold. In this talk, I’ll review this construction and its Riemannian version, which leads to gravitational instantons, and its extension to dimensions greater than 4.