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Thursday, December 1, 2016 2:46

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If the spectrum for the critical value of Kähler coupling strength is complex – say given by the complex zeros of zeta – the preferred extremals of Kähler action are minimal surfaces. This means that they satisfy simultaneously the field equations associated with two variational principles.

**Conservation laws for the minimal surface extremals of Kähler action**

Consider first the basic conservation laws.

- Complex value of α
_{K}means that conserved quantities are complex: this brings strongly in mind twistor approach. The value of cosmological constant is assumed to be real. There are two separate local conservations laws associated with the volume term and Kähler action respectively in both Minkowskian and Euclidian regions. This need not mean separate global conservation laws in Minkowskian and Euclidian regions. If there is non canonical momentum current between Minkowskian (M) and Euclidian (E) space-time regions the real and imaginary parts of conserved quantum numbers correspond schematically to the sums

l Re(Q)= Re(1/α_{K})Q_{K}(E) + Im(1/α_{K})Q_{K}(M) +ρ_{vac} Q_{V}(M)

Im(Q)=Im(1/α_{K})Q_{K}(E) + Re(1/α_{K})Q_{K}(M) .

Here the subscripts _{V} and _{K} refer to the volume term and Kähler action respectively.

Re(Q_{1})= Re(1/α_{K})Q_{K}(E) ,

Re(Q_{2})= Im(1/α_{K})Q_{K}(M) +ρ_{vac} Q_{V}(M) ,

Im(Q_{1})= Im(1/α_{K})Q_{K}(E) ,

Im(Q_{2})= Re(1/α_{K})Q_{K}(M) .

This looks strange and the natural assumption is that canonical momentum currents can flow between the Euclidian and Minkowskian regions and boundary conditions equate the components of normal currents at both sides.

**Are minimal surface extremals of Kähler action holomorphic surfaces in some sense?**

I have proposed several ansätze for the general solutions of the field equations for preferred extremals and the intuitive picture is that the field equations are integrable. Can one make conclusions about general form of solutions.

In D=2 case minimal surfaces are holomorphic surfaces or they hyper-complex variants and the imbedding space coordinates can be expressed as complex-analytic functions of complex coordinate or a hypercomplex analog of this. Field equations stating the vanishing of the trace g^{αβ}H^{k}_{αβ} if the second fundamental form H^{k}_{αβ}equiv D_{α}&partial;_{β}h^{k} are satisfied because the metric is tensor of type (1,1) and second fundamental form of type (2,0) ⊕ (2,0). Field equations reduce to an algebraic identity and functions involved are otherwise arbitrary functions. The constraint comes from the condition that metric is of form (1,1) as holomorphic tensor.

This raises the question whether this finding generalizes to the level of 4-D space-time surfaces and perhaps allows to solve the field equations exactly in coordinates generalizing the hypercomplex coordinates for string world sheet and complex coordinates for the partonic 2-surface.

The known non-vacuum extremals of Kähler action are actually minimal surfaces. The common feature suggested already earlier to be common for all preferred extremals is the existence of generalization of complex structure.

- For Minkowskian regions this structure would correspond to what I have called Hamilton-Jacobi structure. The tangent space of the space-time surface X
^{4}decomposes to local direct sum T(X^{4})= T(X^{2})⊕ T(Y^{2}), where the 2-D tangent places T(X^{2}) and T(Y^{2}) define an integrable distribution integrating to a decomposition X^{4}=X^{2}× Y^{2}. The complex structure is generalized to a direct some of hyper-complex structure in X^{2}meaning that there is a local light-like direction defining light-like coordinate u and its dual v. Y2 has complex complex coordinate (w,wbar). Minkowski space M^{4}has similar structure. It is still an open question whether metric decomposes to a direct sum of orthogonal metrics assignable to X^{2}and Y^{2}or is the most general analog of complex metric in question. g_{uv}and g_{wbar}are certainly non-vanishing components of the induced metric. Metric could allow as non-vanishing components also g_{uw}and g_{vbarw}. This slicing by pairs of surfaces would correspond to decomposition to a product of string world sheet and partonic 2-surface everywhere.

In Euclidian regions ne would have 4-D complex structure with two complex coordinates (z,w) and their conjugates and completely analogous decompositions. In CP_{2} one has similar complex structure and actually Kähler structure extending to quaternionic structure. I have actually proposed that quaternion analyticity could provide the general solution of field equations.

A=J^{α}_{γ} J^{γβ}H^{k}_{αβ} .

Second term comes from the variation with respect to induced Kähler form and is proportional to

B=j^{α} P^{k}_{s}J^{s}_{l}∂_{α}h^{l} .

Here P^{k}_{l} is projector to the normal space of space-time surface and j^{α}= D_{β} J^{αβ} is the conserved Kähler current.

For the known extremals j vanishes or is light-like (for massless extremals) in which case A and B vanish separately.

See the new chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? or the article with the same title.

For a summary of earlier postings see Latest progress in TGD.