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

TL;DR

This paper reveals a deep connection between stationary axisymmetric black hole solutions in four-dimensional vacuum Einstein gravity and self-dual Yang-Mills fields in 2+2 dimensions, showing black holes can serve as backgrounds for N=2 superstrings.

Contribution

It establishes a novel correspondence between black hole solutions and self-dual gauge fields, enabling black holes to act as consistent backgrounds for N=2 superstring theory.

Findings

Black hole solutions correspond to self-dual Yang-Mills solutions in 2+2 dimensions.

Physical parameters of black holes relate to instanton-like invariants.

Kerr solutions with naked singularities may be interpreted as closed string backgrounds.

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.