**Authors:** Matti Pitkänen

There are three separate approaches to the challenge of constructing WCW Kähler geometry
and spinor structure. The first one relies on a direct guess of Kähler function. Second approach
relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry
needed to guarantee the mathematical existence of Riemann connection. The third approach relies
on the construction of spinor structure assuming that complexified WCW gamma matrices are
representable as linear combinations of fermionic oscillator operator for the second quantized free
spinor fields at space-time surface and on the geometrization of super-conformal symmetries in
terms of spinor structure.
In this article the construction of Kähler form and metric based on symmetries is discussed.
The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with
lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy
ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality
and the basic objects are pairs partonic 2-surfaces X^{2} at opposite light-like boundaries of causal
diamonds (CDs).
The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric
spaces G/H labeled by zero modes having an interpretation as classical, non-quantum
uctuating
variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary
δM_{+}^{4}
+ and of light-like 3-surfaces implying a generalization of conformal invariance. The group
G acting as isometries of WCW is tentatively identified as the symplectic group of
δM_{+}^{4} x CP_{2}
localized with respect to X^{2}. H is identified as Kac-Moody type group associated with isometries
of H = M_{+}^{4} x CP_{2} acting on light-like 3-surfaces and thus on X^{2}.
An explicit construction for the Hamiltonians of WCW isometry algebra as so called
ux
Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of
these. Explicit expressions for WCW
ux Hamiltonians as functionals of complex coordinates of
the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic
2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are
proposed.

**Comments:** 26 Pages.

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[v1] 16 Jun 2010

[v2] 2012-01-30 22:03:21

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