Stationary Axisymmetric Black Holes, $N=2$ Superstring, and Self-Dual Gauge or Gravity Fields

Abstract

We present interesting relationship between what are called stationary axisymmetric black hole solutions for the vacuum Einstein equations in the ordinary four-dimensions and exact solutions for self-dual Yang-Mills fields in flat $2+2$ dimensions which are nothing but the consistent backgrounds for $~N=2$~ open superstring. We show that any stationary axisymmetric black hole solution for the former automatically provides an exact solution for the latter. We also give a nice relation between the physical parameters of black holes and an invariant integral analogous to instanton charges for such a self-dual Yang-Mills solution. This result indicates that any general black hole solution in the usual four-dimensions can be the background of $~N=2$~ superstring at the same time. We also give an interesting embedding of stationary axisymmetric solutions into the background of $~N=2$~ closed superstring. Finally we show some indication that the Kerr solution with naked singularity (for $a > m$) is nothing else than the gravitational field generated by a closed string.