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 Lind and Newman . Combination of these strings forms a membrane source of the Kerr geometry which is parallel to the string/M-theory unification. 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 critical N=2 superstring.


Introduction
In section 4. we show that the source of the complex Kerr geometry is to be an open complex string. It is closely linked with the old remarks by Ooguri and Vafa, that the complex world lines (CWL) parametrized by complex time parameter τ = t + iσ, are in fact to be world-sheets of the complex string, [26]. It has been shown [23] that the boundary conditions of the complex Kerr string, generating the complex Kerr geometry, require orientifold structure of its world-sheet.
Finally, we observe that the structure of the membrane (vacuum bubble [11]) source of the real Kerr geometry turns out to be parallel to formation of the membrane in the superstring/M-theory unification. Namely, the closed Kerr string of the real Kerr geometry grows by extra world-sheet parameter of the open complex Kerr string, [27], resulting in formation of the Kerr bubble-membrane source, which is parallel to construction of enhancon in string/M-theory [28].
This parallelism of the structure of Kerr geometry with basic structures of the superstring theory, and in particular, the inherent existence of the K3 surface, enforces us to draw a parallel with the critical N=2 superstring theory [38], which describes a complex two-dimensional string (four real dimensions) and is closely related with twistor theory too. This allows us to suggest that the complex Kerr string represents an embedding of the critical N=2 superstring theory into complex Kerr geometry.
2 Real structure of the KN geometry KN metric is represented in the Kerr-Schild (KS) form [14], where η µν is auxiliary Minkowski background in Cartesian coordinates x = x µ = (t, x, y, z), and e 3 (x) is a tangent direction to a Principal Null Congruence (PNC), which is determined by the form 2 via function Y (x), which is obtained from the Kerr theorem, [14,29,30,31,32,33].
The PNC forms a caustic at the Kerr singular ring, r = cos θ = 0. As a result, the KN metric (1) and electromagnetic potential are aligned with Kerr PNC and concentrate near the Kerr ring, forming a closed stringwaveguide for traveling electromagnetic waves [2,3,18]. Analysis of the Kerr-Sen solution to low energy string theory [5] showed that similarity of the Kerr ring with a closed strings is not only analogue, but it has really the structure of a fundamental heterotic string [4]. Along with this closed string, the KN geometry contains also a complex open string, [23], which appears in the initiated by Newman complex representation of Kerr geometry, [25]. This string gives an extra dimension θ to the stringy source (θ ∈ [0, π]), resulting in its extension to a membrane (bubble source [9,11]. A superstring counterpart of this extension is a transfer from superstring theory to 11-dimensional M -theory and M 2-brane, [27]. Kerr Theorem determines the shear free null congruences with tangent direction (3) by means of the solution Y (x) of the equation where F (T A ) is an arbitrary holomorphic function in the projective twistor space with CP 3 coordinates Using the Cartesian coordinates x µ , one can rearrange variables and reduce function F (T A ) to the form F (Y, x µ ), which allows one to get solution of the equation (5) For the Kerr and KN solutions, the function F (Y, x µ ) turns out to be quadratic in Y, and the equation (5) represents a quadric in the projective twistor space CP 3 , with a non-degenerate determinant ∆ = (B 2 − 4AC) 1/2 which determines the complex radial distance [32,34] This case is explicitly resolved and yields two solutions which allows one to restore two PNC by means of (3).
One can easily obtain from (7) and (9) that the used in the metric (2) and the em potential (4) complex radial distancer = r + ia cos θ may also be determined from the Kerr generating function by the relationr Therefore, the Kerr singular ring,r = 0, is formed as a caustic of the Kerr congruence, As a consequence of Vieta's formulas, the quadratic in Y function (7) may be expressed via the roots Y ± (x µ ) in the simple form

Complex Kerr geometry and the complex retarded-time construction
KN solution was initially obtained in [12] by a "complex trick" from the Kerr solution.
One can see that the complex radial distancer = r+ia cos θ takes in Cartesian coordinates the formr = and therefore, the scalar component of the vector potential (4) may be obtained from the Coulomb solution φ( x) = e/r = e/ x 2 + y 2 + z 2 by a complex shift z → z + ia, or by the shift of its singular point x 0 = (0, 0, 0) in complex region x 0 → (0, 0, −ia). This shift was first described in 1887 by Appel [24], who noticed that the Coulomb solution, being invariant solution to the linear Laplace equation by the real shifts x → x+ a, should also be invariant for the complex shift. In spite of triviality of this procedure from complex point of view, it yields very nontrivial consequences in the real section, in particular, the singular point of the Coulomb solution x 0 = (0, 0, 0) turns into singular ring x 2 + y 2 + (z + ia) 2 = 0, intersection of the sphere x 2 + y 2 + z 2 = a 2 and plane z = 0, and the space turns out to be twosheeted, branching around this singular ring.
Lind and Newman showed, [25], that the linearized KN solution corresponds to this complex shift and may be generated by a complex source propagating along a complex world line, and suggested a special complex retarded-time procedure which generalizes the usual real retarded-time construction. It has been shown later, [32,34], that the complex retarded-time representation is exact, if the KN solution is presented in the Kerr-Schild form. Therefore, the exact KN solution may be described as a field generated by a complex source propagating along complex world-line where u µ = (1, 0, 0, 0), k R = (1, 0, 0, −1), k L = (1, 0, 0, 1). Index L labels it as a Left structure, and we should add a complex conjugate Right structure Therefore, from complex point of view the Kerr and Schwarzschild geometries are equivalent and differ only by their real slice, which for the Kerr solution goes aside of its center. Complex shift turns the Schwarzschild radial directions n = r/|r| into twisted directions of the Kerr congruence, Fig.1.

Complex Kerr's string
It was obtained [23,26] that the complex world line x µ 0 (τ ), parametrized by complex time τ = t + iσ, represents really a two-dimensional surface which takes an intermediate position between particle and string. The corresponding "hyperbolic string" equation [26], ∂ τ ∂τ x 0 (t, σ) = 0, yields the general solution as sum of the analytic and anti-analytic modes x L (τ ), x R (τ ), which are not necessarily complex conjugate. For each real point x µ , the parameters τ andτ should be determined by a complex retarded-time construction. Complex source of the KN solution corresponds to two straight complex conjugate world-lines, (14), (15). Contrary to the real case, the complex retarded and advanced times τ ∓ = t ∓r may be determined by two different (Left or Right) complex null planes, which are generators of the complex light cone. It yields four different roots for the Left and Right complex structures [32,34] τ ∓ The real slice condition determines relation σ = a cos θ with null directions of the Kerr congruence θ ∈ [0, π], which puts restriction σ ∈ [−a, a] indicating that the complex string is open, and its endpoints σ = ±a may be associated with the Chan-Paton charges of a quark-antiquark pair. In the real slice, the complex endpoints of the string are mapped to the north and south twistor null lines, θ = 0, π, see Fig.3. Orientifold. The complex open string boundary conditions [23] require the worldsheet orientifold structure [27,35,36,37,38] which turns the open string in a closed but folded one. The world-sheet parity transformation Ω : σ → −σ reverses orientation of the world sheet, and covers it second time in mirror direction. Simultaneously, the Left and Right modes are exchanged. 3 The projection Ω is combined with space reflection R : r → −r, resulting in RΩ :r → −r, which relates the retarded and advanced folds preserving analyticity of the world-sheet. The string modes x L (τ ), x R (τ ), are extended on the second half-cycle by the well known extrapolation, [27,38] x which forms the folded string, in which the retarded and advanced modes are exchanged every half-cycle. The real KN solution is generated by the straight complex world line (CWL) (14) and by its conjugate Right counterpart (15). By excitations of the complex string, the orientifold condition (20) becomes inconsistent with the complex conjugation of the string ends, and the world lines x L (τ ), and x R (τ ) should represent independent complex sources. The projection T = RΩ sets parity between the positive Kerr sheet determined by the Right retarded time and the negative sheet of the the Left advanced time. It allows one to escape the anti-analytical Right complex structure, replacing it by the Left advanced one, and the problem is reduced to self-interaction of the retarded and advanced sources determined by the time parameters τ ± . For any non-trivial (not straight) CWL, the Kerr theorem will generate different congruences for τ + , and τ − . Each of these sources produces a twosheeted Kerr-Schild geometry, and the formal description of the resulting four-folded congruence should be based on the multi-particle Kerr-Schild solutions, [33]. 4 The corresponding two-particle generating function of the Kerr theorem will be where F L and F R are determined by x L (τ + ), and x L (τ − ). The both factors are quadratic in T A . The corresponding equation describes a quartic in CP 3 which is the well known Calabi-Yau two-fold, [27,38]. We arrive at the result that excitations of the Kerr complex string generate a Calabi-Yau two-fold, or K3 surface, on the projective twistor space CP 3 .

Outlook.
One sees that the Kerr-Schild geometry displays striking parallelism with basic structures of superstring theory. However, our principal result in this paper is the presence of inherent Calabi-Yau twofold in the complex twistorial structure of the Kerr geometry. In the recent paper [18] we argued that it is not accidental, because gravity is a fundamental part of the superstring theory. However the Kerr-Schild gravity, being based on twistor theory, displays also some inherent relationships with superstring theory. In many respects the Kerr-Schild gravity resembles the twistor-string theory, [40,41,21], which is also four-dimensional, based on twistors and related with experimental particle physics. On the other hand, the complex Kerr string has mach in common with the N=2 superstring [26,38,43]. It is also related with twistors and has the complex critical dimension two which corresponds to four real dimensions and indicated that N=2 superstring may lead to four-dimensions. However, signature of the N=2 string may only be (2,2) or (4,0), which caused the obstacles for embedding of this string in the space-times with minkowskian signature. Up to our knowledge, this trouble was not resolved so far, and the initially enormous interest to N=2 string seems to be dampened. Meanwhile, embedding of the N=2 string in the complexified Kerr geometry is almost trivial task. It hints that stringlike structures of the real and complex Kerr geometry are not simply analogues, but reflect the underlying dynamics of the N=2 superstring theory, In the same time, along with wonderful parallelism, the stringy system of the fourdimensional KN geometry displays very essential peculiarities, which make it closer to particle physics.
• The spin/mass ratio of the spinning particles is extremely high which leads to the over-rotating BH geometry without horizons and offers a new application, next to traditional attention of superstring theory to quantum black holes.
• The supplementary Kaluza-Klein space is absent, and the role of compactification circle is played by the naked Kerr singular ring with traveling waves, which realizes a "compactification without compactification", [18].
• The lightlike twistorial rays are tangent to the Kerr singular ring, indicating that the Kerr ring is the lightlike string, and it may play the role of DLCQ circle of M-theory, [42].
• Consistency of the KN solution with gravitational background of the electron, [8,9,13,18], shows that the 4d Kerr characteristic length of the Kerr ring, a = J/m, corresponds to the Compton scale of spinning particles.
The considered stringy structures of the real and complex Kerr geometry set a parallelism between the 4d Kerr geometry and superstring theory, indicating that complexification of the Kerr geometry may serve an alternative to traditional compactification of higher dimensions.