There are pressing motivations for understanding the preferred extremals of Kähler action. For instance, the conformal invariance of string models naturally generalizes to 4-D invariance defined by quantum Yangian of quantum affine algebra (Kac-Moody type algebra) characterized by two complex coordinates and therefore explaining naturally the effective 2-dimensionality (see this). The problem is however how to assign a complex coordinate with the string world sheet having Minkowskian signature of metric. One can hope that the understanding of preferred extremals could allow to identify two preferred complex coordinates whose existence is also suggested by number theoretical vision giving preferred role for the rational points of partonic 2-surfaces in preferred coordinates. The best one could hope is a general solution of field equations in accordance with the hints that TGD is integrable quantum theory.
A lot is is known about properties of preferred extremals and just by trying to integrate all this understanding, one might gain new visions. The problem is that all these arguments are heuristic and rely heavily on physical intuition. The following considerations relate to the space-time regions having Minkowskian signature of the induced metric. The attempt to generalize the construction also to Euclidian regions could be very rewarding. Only a humble attempt to combine various ideas to a more coherent picture is in question.
The core observations and visions are following.
- Hamilton-Jacobi coordinates for M4 (discussed in this chapter) define natural preferred coordinates for Minkowskian space-time sheet and might allow to identify string world sheets for X4 as those for M4. Hamilton-Jacobi coordinates consist
of light-like coordinate m and its dual defining local 2-plane M2⊂ M4 and complex transversal complex coordinates (w,w*) for a plane E2x orthogonal to M2x at each point of M4. Clearly, hyper-complex analyticity and complex analyticity are in question.
- Space-time sheets allow a slicing by string world sheets (partonic 2-surfaces) labelled by partonic 2-surfaces (string world sheets).
- The quaternionic planes of octonion space containing preferred hyper-complex plane are labelled by CP2, which might be called CP2mod (see this). The identification CP2=CP2mod motivates the notion of M8--M4× CP2 duality (see this). It also inspires a concrete solution ansatz assuming the equivalence of two different identifications of the quaternionic tangent space of the space-time sheet and implying that string world sheets can be regarded as strings in the 6-D coset space G2/SU(3). The group G2 of octonion automorphisms has already earlier appeared in TGD framework.
- The duality between partonic 2-surfaces and string world sheets in turn suggests that the CP2=CP2mod conditions reduce to string model for partonic 2-surfaces in CP2=SU(3)/U(2). String model in both cases could mean just hypercomplex/complex analyticity for the coordinates of the coset space as functions of hyper-complex/complex coordinate of string world sheet/partonic 2-surface.
The considerations of this section lead to a revival of an old very ambitious and very romantic number theoretic idea, which I already thought to be dead.
- To begin with express octonions in the form o=q1+Iq2, where qi is quaternion and I is an octonionic imaginary unit in the complement of fixed a quaternionic sub-space of octonions. Map the preferred coordinates of H=M4× CP2 to octonionic coordinate, form an arbitrary octonion analytic function having expansion with real Taylor or Laurent coefficients to avoid problems due to non-commutativity and non-associativity. Map the outcome to a point of H to get a map H→ H. This procedure is nothing but a generalization of Wick rotation to get an 8-D generalization of analytic map.
- Identify the preferred extremals of Kähler action as surfaces obtained by requiring the vanishing of the imaginary part of an octonion analytic function. Partonic 2-surfaces and string world sheets would correspond to commutative sub-manifolds of the space-time surface and of imbedding space and would emerge naturally. The ends of braid strands at partonic 2-surface would naturally correspond to the poles of the octonion analytic functions. This would mean a huge generalization of conformal invariance of string models to octonionic conformal invariance and an exact solution of the field equations of TGD and presumably of quantum TGD itself.
For background see the chapter TGD as Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts and the article An attempt to understand preferred extremals of Kauml;hler action