Monday, February 14, 2005

Kähler calibrations, number theoretical compactification, and general solution to the absolute minimization of Kähler action

The title probably does not say much to anyone who is not a theoretical physicist working with theories of everything. I thought however it appropriate to glue this piece of text from my homepage in hope that the information about these beautiful discoveries might find some readers. So what follows is rather heavy mathematical jargon. Calibrations represent a good example of those merciful "accidents", which happen just the right time. Just for curiosity I decided to look what the word means, and it soon became clear that the notion of calibration allows to formulate my proposal for how to construct general solution of field equations defined by Kähler action in terms of a number theoretic spontaneous compactification in a rigorous and..., perhaps I dare say it aloud, even in convincing manner. For an excellent popular representation about calibrations, spinors and super-symmetries see the homepage of Jose Figueroa-O'Farrill . 1. The notion of calibration Calibrations allow a very elegant and powerful formulation of minimal surface property and have been applied also in brane-worldish considerations. Calibration is a closed p-form, whose value for a given p-plane is not larger than its volume in induced metric. What is important that if it is maximum for tangent planes of p-sub-manifold, minimal surface with smallest volume in its homology equivalence class results. Could absolute minima of Kähler action found using Kähler calibration?! For instance, all surfaces X^2xY^2 subset M^4xCP_2, X^2 and Y^2 minimal surfaces, are solutions of field equations. Calibration theory allows to concluded that Y^2 is any complex manifold of CP_2! A very general solution of TGD in stringy sector results and there exists a deformation theory of calibrations to produce moduli spaces for the perturbations of these solutions! In fact, all known solutions of field equations are either minimal surfaces or have a vanishing Kähler action density. This probably tells more about my simple mind-set than reality, and there are excellent reasons to believe that, since Lorentz-Kähler force vanishes, the known solutions are space-time correlates for asymptotic self-organization patterns. The question is how to find more general solutions. Or how to generalize the notion of calibration for minimal surfaces to what might be called Kähler calibration? It is here, where the handsome and young idea of number theoretical spontaneous compactification enters the stage and the outcome is a happy marriage of two ideas. 2. The notion of Kähler calibration It is intuitively clear that the closed calibration form omega which is saturated for minimal surfaces must be replaced by Kähler calibration 4-form omega_K= L_K omega . L_K is Kähler action density (Maxwell action for induced CP_2 Kähler form).

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.
    Conclusion: M^8 is mapped to M^4xCP_2 and metric of H and CP_2 Kähler form can be induced. M^8 having originally only the number theoretical norm as metric inherits the metric of H.
  • 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 omega_K= L_K*omega

    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).
4. Could TGD reduce to string model like theory in HO picture? Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the local automorphism associated with the hyper-octonionic spinor field is dictated by field equations of some kind. The experience with WZW model suggests that in case of non-vacuum extremals G_2 element could be written as a product g=g_L(h)g^{-1}_R(h*) of hyper-octonion analytic and anti-analytic complexified G_2 elements. g would be determined by the data at hyper-complex 2-surface for which the tangent space at a given point is spanned by real unit and preferred hyper-octonionic unit. Also Dirac action would be naturally restricted to this surface. The amazing possibility is that TGD could reduce in HO picture to 8-D WZW string model both classically and quantally since vertices would reduce to integrals over 1-D curves. The interpretation of generalized Feynman diagrams in terms of generalized braid/ribbon diagrams and the unique properties of G_2 provide further support for this picture. In particular, G_2 is the lowest-dimensional Lie group allowing to realize full-powered topological quantum computation based on generalized braid diagrams and using the lowest level k=1 Kac Moody representation. Even if this reduction would occur only in special cases, such as asymptotic solutions for which Lorentz Kähler force vanishes or maxima of Kähler function, it would mean enormous simplification of the theory. 5. Why extermals of Kähler action would correspond to hyper-quaternionic 4-surfaces? The resulting over all picture leads also to a considerable understanding concerning the basic questions why (co)-hyper-quaternionic 4-surfaces define extrema of Kähler action and why WZW strings would provide a dual for the description using Kähler action. The answer boils down to the realization that the extrema of Kähler action minimize complexity, also algebraic complexity, in particular non-commutativity. A measure for non-commutativity with a fixed preferred hyper-octonionic imaginary unit is provided by the commutator of 3 and 3bar parts of the hyper-octonion spinor field defining an antisymmetric tensor in color octet representation: very much like color gauge field. Color action is a natural measure for the non-commutativity minimized when the tangent space algebra closes to complexified quaternionic, instead of complexified octonionic, algebra. On the other hand, Kähler action is nothing but color action for classical color gauge field defined by projections of color Killing vector fields. Here it is!That WZW + Dirac action for hyper-octonionic strings would correspond to Kähler action would in turn be the TGD counterpart for the proposed string-YM dualities. 6. Summary To sum up, the following conjectures are direct generalizations of those for minimal surfaces.
  • 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!
For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.

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