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Complexification as alternative to higher dimensions: N = 2 string as a complex source of Kerr geometry.
Alexander Burinskii NSI, Russian Academy of Sciences, Moscow, Russia 16th Lomonosov Conference on Elementary Particle Physics. Moscow State University, 26 August, 2013. Based on: A.B., Stringlike structures in the Kerr-Schild geometry: N=2 string, twistors and Calabi-Yau twofold [arXiv:1307.5021 ]. A.B., Complex Structure of the Four-Dimensional Kerr Geometry Adv. High Energy Phys. v.2013, ID 509749, [arXiv:1211.6021 ]. A.B., String-like Structures in Complex Kerr Geometry, [arXiv:gr-qc/9303003]

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BLACK HOLES - STRINGS - PARTICLES
STRING THEORY: "... realistic model of elementary particles stil l appears to be a distant dream." (J. Schwarz, arXiv:1201.0981 ) KERR GEOMETRY corresp onds to background of an electron! Measurable parameters of an electron (mass, spin, charge, magnetic moment) indicate that its gravitational and electromagnetic field correspond to Kerr-Newman solution.(Carter 1968, Israel 1970, AB 1974, L´ ez 1984,...et op
al.)

Kerr's gravity as a BRIDGE: Spinning Particles Kerr's Gravity String theory SPIN of particles is extreme high: over-rotating geometry without horizons! a / m = 1044 NAKED SINGULAR RING, which was interpreted as closed string ( Ivanenko& AB 1975 ). Fundamental string solutions to low-energy string theory ( Witten 1985, Horowitz & Steif 1990, Sen 1992, A. Tseytlin 1993, AB 1995.) Strings as Solitons & Black Holes as Strings, (Dabholkar at al 1995).

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KERR's STRINGY SYSTEM
Second complex string app ears in complex structure of Kerr geometry, (AB 1993). TWISTOR STRUCTURE OF THE KERR GEOMETRY. Inherent CalabiYau space appears as a quartic in the projective twistor space C P 3 . The closed Kerr string and op en complex string form together 4D stringmembrane system , which is parallel with string/M-theory unification (AB, arXiv:1211.6021). PROPOSITION: Emergence of this similarity is the N = 2 sup erstring, structure of which is remarkable similar to structure of COMPLEX SOURCE OF KERR GEOMETRY! Recently, these stringy structures were indep endently discussed by Adamo and Newman: "...It would have been a cruel god to have layed down such a pretty scheme and not have it mean something deep." (Adamo&Newman,
PRD 2011).

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N=2 sup erstring is one of the three consistent critical string theories: D=4, D=10, and D=26. N=2 string (D=4) is complex. M.Geen, J.Schwarz and E.Witten, Superstring Theory V.1. "...N = 2 extension of the superstring construction gives a highly symmetric two-dimensional (complex) theory an interesting generalization of the super-Virasoro algebra. It seemingly cannot be given the usual interpretation of a string theory... Perhaps it enters physics in some other and yet unknown way... ... crucial subtleties in this theory have not yet been unraveled."

Z µ = X µ + iY µ,

µ = 0, 1 1 ¯ ¯ d2 {Z Z - i } S=- 2

The global N = 2 sup ergauge transformations Z = ¯ , = -i Z. "...there are no transverse oscil lations at al l... the massless scalar ground state is the only propagating degree of freedom...(at least for this sector). However, subtleties in the quantization... have been pointed out recently, and this statement may require revision."
"Subtleties" of the N=2 string could be connected with problem of boundary conditions which requires orientifold projection (invented later by L.Dixon,J.A.Harvey, C.Vafa, and E.Witten, Nucl.Phys.1987).
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Complex Structure of the Kerr geometry. Complex shift. Appel 1887!
Complex shift of the Coulomb solution (x) in Cartesian coordinates x = (x, y , z ) Ta(x) = a(x + i~), where ~ = (0, 0, -ia), creates in the real slice Appel solution a = Re ~ ~a a in oblate spheroidal coordinates r and .
3

e r+ia cos

,

Z
2

=const.

1

r>0

0

r=const
-1

-2

-3 -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

· Complex shift Ta : (x, y , z ) (x, y , z - ia) of the Schwarzschild solution in the KerrSchild form gµ = µ + 2H kµk , creates in the real slice the Kerr solution. · Complex shift Ta : (x, y , z ) (x, y , z - ia) of the Reisner-Nordstr¨ solution in the om Kerr-Schild form gµ = µ + 2H kµk , creates the Kerr-Newman solution. · Global N=2 super-translation S Ta, : of the Kerr-Newman solution in the Kerr-Schild form creates in the real slice the Super-Kerr-Newman solution to broken N=2 Supergravity (AB, arXiv:hep-th/9903032). Trivial operation of the shift is combined with nonlinear op eration of the real slice.
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Complex retarded-time construction. (Lind & Newman, 1974) Kerr's source is considered as a mysterious "particle" propagating along a complex world-line xµ( ) parametrized by complex time = t + i . 0 System of the complex light cones emanating by complex world-line (xµ - x0µ)(xµ - xµ) = 0 splits into families of the "Left" and "Right" com0 plex null planes: Ai ~A KL = {x : x = xi (L) + L AAR }. (1) 0 ~ ~ "Left" (L =const; R -var.) and "Right"(R =const; L -var.). The Kerr congruence K emerges as real slice of the "Left" null planes (Y = const.). Complex Kerr String. (AB, gr-qc/9303003) Complex world line xµ(t + i ) is really a world sheet of a complex string 0 parametrized by t and . Real slice fixes = a cos [-a, a]. String is op en with the end p oints = ±a. Boundary conditions require orientifolding the world-sheet. The orientifold parity - reverses orientation of the world sheet, and covers it second time in mirror direction. Two oriented copies of the interval = [-a, a], are joined, forming world-sheet of a closed but folded string.
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REAL structure of the Kerr-Newman solution: Metric
mr - e2/2 gµ = µ + 2Hkµk , H = 2 , (2) r + a2 cos2 and electromagnetic vector potential is Aµ = Re r+iaecos kµ. The Kerr singular ring is a KN
Z
10

5

0

-5

-10 10 5 0 0 -5 -10 -10 -5 5 10

branch line forming TWOSHEETED Kerr space!

Kerr congruence is controlled by KERR THEOREM: as analytic solution of the equation F (T a) = 0 , where F is a holomorphic function of the projective twistor coordi¯ nates Ta = {Y, - Yv, u + Y } CP3. For the Kerr-Newman solution function F is quadratic in Y, which yields TWO roots Y±(x) resulting in twosheeted Kerr background! Functions F(Ta) of higher degrees in Y correspond to multi-sheeted geometry and multiparticle solutions, [AB (2006)]. Orientifold doubles the number of Kerr's sheets, which is described by quartic eq. in CP3, creating inherent Calabi -Yau twofold.
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z+ string
singular ring
4

2

= const.

0

-2

-4

-6

-8 2 1 0 -6

real slice of complex string

-1 -2 -3

z- string
3 2 1 0 -1

-2

-3

-4

-5

Figure 1: Four ro ots for the retarded and adv r adv r advanced times, XL , XLet and XR , XRet creating the K3 surface.

Figure 2: One sheet of the K3 for r > 0 and = const. Kerr congruence is tangent to singular ring at = /2.

The real closed string and complex Kerr string form together a 4D stringmembrane system is parallel with string/M-theory unification (AB, arXiv:1211.6021) The inherent Calabi-Yau space (K3 surface) is pro jected into real 4D Kerr geometry in the form of analytic extension of the Kerr principal null congruence.

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Problem of emb edding of the N = 2 string was principal obstacle for its application: embedding in the real minkowskian space-time is only consistent with signatures (2,2) or (4,0). There is no problem for emb edding in the complex 4D Kerr geometry, since diverse sections may have different signatures. Kerr's complex source and sup ersymmetry? "...Wess-Zumino formalism may have close connection with the Twistor formalism of Penrose..." (SalamStrathdee, 1974) Fermionic part of the N = 2 superstring (the Dirac spinor) plays important role fixing the Left null planes of twistorial structure of the complexified 4d Kerr geometry. Conclusion. I. N = 2 sup erstring may consistently b e emb edded in the complex 4D Kerr geometry, playing the role of its complex source. I I. Complexification is alternative to higher dimensions. I I I. Planck scale is replaced by Compton scale of Kerr geometry.

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THANK YOU FOR ATTENTION!

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