Stringlike structures in Kerr–Schild geometry: The $N{=}2$ string, twistors, and the Calabi–Yau twofold

The four-dimensional Kerr–Schild geometry contains two stringy structures. The first is the closed string formed by the Kerr singular ring, and the second is an open complex string obtained in the complex structure of the Kerr–Schild geometry. The real and complex Kerr strings together form a membrane source of the over-rotating Kerr–Newman solution without a horizon, $a=J/m\gg m$. It was also recently found that the principal null congruence of the Kerr geometry is determined by the Kerr theorem as a quartic in the projective twistor space, which corresponds to an embedding of the Calabi–Yau twofold into the bulk of the Kerr geometry. We describe this embedding in detail and show that the four sheets of the twistorial K3 surface represent an analytic extension of the Kerr congruence created by antipodal involution.