Important point: omega_K is closed but not omega as in the case of minimal surfaces. L_K acts as an integrating factor. This difference is absolutely essential. When L_K is constant you should get minimal surfaces. L_K is indeed constant for the known minimal surface solutions. The basic objection against this conjecture is following: L_K is a four dimensional action density. How it is possible to assign it to a 4-form in 8 dimensional space-time? Here number theoretical spontaneous compactification shows its power. 3. Number theoretic compactification allows to define Kähler calibration The calibrations are closely related to spinors and the number theoretic compactification based on 2-component octonionic spinors satisfying Weyl condition, and therefore equivalent with octonions themselves, tells how to construct omega.
- The hyper-octonion real-analytic maps of HO=M^8 to itself define octonionic 2 spinors satisfying Weyl condition. Octonionic massless Dirac equation reduces to d'Alembert equation in M^8 by the generalization of Cauchy-Riemann conditions.
- Octonions and thus also the spinors have 1+1+3+3bar decomposition with respect to (color) SU(3) sub-group of octonion automorphism G_2. SU(3) leaves a preferred hyper-octonionic imaginary unit invariant. The unit can be chosen in local manner and the choices are parameterized by local S^6.
- 3x3bar tensor product defines color octet identifiable as SU(3) Lie algebra generator and its exponentiation gives SU(3) group element.
- The canonical bundle projection SU(3)-->CP_2 assigns a CP_2 point to each point of M^8, when a preferred octonion unit is fixed at each point of M^8.
- Canonical projection M^8-->M^4 assigns M^4 point to point of M^8.
- Here comes the key point of the construction. CP_2 parameterizes hyper-quaternionic planes of hyper-octonions and therefore it is possible to assign to a given point of M^8 a unique hyper-quaternion 4-plane. Thus also the projection J of Kähler form to this plane and also the dual *J of this projection. Therefore also L_K=J\wedge*J as the value of Kähler action density!
- The Kähler calibration
is defined in an obvious manner. As found, L_K is associated with the local hyper-quaternionic plane is assigned to each point of M^8. The form omega is obtained from the wedge product of unit tangent vectors for hyper-quaternionic plane at a given point by lowering the indices using the induced metric in M^8. omega is not a closed form in general. For a given 4-plane it is essentially the cosine of the angle between plane and hyper-quaternionic plane and saturated for hyper-quaternionic plane so that calibration results.
- Kähler calibration is the only calibration that one can seriously imagine. Furthermore, the spinorial expression for omega is well defined only if the form omega saturates for hyper-quaternionic planes or their duals. The reason is that non-associativity makes the spinorial expression involving an octonionic product of four tangent vectors for the calibration ill defined for non-associative 4-planes. Hence number theory allows only hyper-quaternionic saturation. Note that also co-hyper-quaternionicity is allowed and required by the known extremals of Kähler action. A 4-parameter foliation of M^8, and perhaps even that of M^4xCP_2 (discrete set of intersections probably occurs) by 4-surfaces results and the parameters at given point of X^4 define the dual space-time surface.
- A surprise, which does not flatter theoretician's vanity, emerges. Closed-ness of omega_K implies that if absolute value of Kähler action density replaces K\"ahler action, minimization indeed occurs for hyper-quaternionic surfaces in a given homology class assuming that the hyper-quaternionic plane at given point minimizes L_K (is this equivalent this closed-ness of omega_K?). Thus L_K should be replaced with L_K so that vacuum extremals become absolute minima, and universe would do its best to save energy by staying as near as possible to vacuum. The 3 surfaces for which CP_2 projection is at least 2-dimensional and not Lagrange manifolds would correspond to non-vacua since since conservation laws do not leave any other option. The attractiveness of this option from the point of calculability of TGD would be that the initial values for the time derivatives of the imbedding space coordinates at X^3 at light-like 7-D causal determinant could be computed by requiring that the energy of the solution is minimized. This could mean a computerizable solution to the absolute minimization.
- There is a very beautiful connection with super-symmetries allowing to express absolute minimum property as a condition involving only the hyper-octonionic spinor field defining the Kähler calibration (discovered for calibrations by Strominger and Becker).
- L_K acts as an integrating factor and omega_K= L_K*omega is a closed form.
- Generalizing from the case of minimal surfaces, closed-ness guarantees that hyper-quaternionic 4-surfaces saturating this form are absolute minima of Kähler action.
- The hyper-octonion analytic solutions of hyper-octonionic Dirac equation defines those maps M^8-->M^4xCP_2 for which L_K acts as an integrating factor. Classical TGD reduces to a free Dirac equation for hyper-octonionic spinors!