Complex Structure of the Four-Dimensional Kerr Geometry: Stringy System, Kerr Theorem, and Calabi-Yau Twofold

The 4D Kerr geometry displays many wonderful relations with quantum world and, in particular, with superstring theory. The lightlike structure of fields near the Kerr singular ring is similar to the structure of Sen solution for a closed heterotic string. Another string, open and complex, appears in the complex representation of the Kerr geometry initiated by Newman. Combination of these strings forms a membrane source of the Kerr geometry which is parallel to the structure of M-theory. In this paper we give one more evidence of this relationship, emergence of the Calabi-Yau twofold (K3 surface) in twistorial structure of the Kerr geometry as a consequence of the Kerr theorem. Finally, we indicate that the Kerr stringy system may correspond to a complex embedding of the criticalN=2superstring.

THE KERR THEOREM AND MULTI-PARTICLE KERR–SCHILD SOLUTIONS

We discuss an extended version of the Kerr theorem which allows one to construct the exact solutions of the Einstein–Maxwell field equations from a holomorphic generating function F of twistor variables. The exact multi-particle Kerr–Schild solutions are obtained from generating function of the form [Formula: see text], where Fi are partial generating functions for the spinning particles i = 1 ⋯ k. Solutions have an unusual multi-sheeted structure. Twistorial structures of the ith and jth particles do not feel each other, forming a type of its internal space. Gravitational and electromagnetic interaction of the particles occurs via the light-like singular twistor lines. As a result, each particle turns out to be "dressed" by singular pp-strings connecting it to other particles. We argue that this solution may have a relation to quantum theory.