Wednesday, September 22, 2010

A comment about formulation of TGD in the product of twistor space and its dual

In previous postings (see this, this, this, and this: see also short file and the pdf article at my homepage) I have developed crazy ideas about the generalization of the Grassmannian-twistor program relaying on Yangian symmetry program to TGD.

One of the latest steps forward was the realization of almost trivial fact that one can obtain the points of M4× CP2 by a canonical mapping twistors to its dual so that each twistor defines a complex plane CP2 the dual space. Conformally compactified M4 is in turn obtained as lines of CP3. This inspired the vision that TGD could be formulated in the product CP3× CP3 of twistor spaces with the factors endowed with conformal metric with signature (2,4) and with metric having Euclidian signature respectively. Ordinary CP3 is Calabi-Yau manifold and one can hope that this is true in generalized sense for its ultra-Minkowskian dual.

The fascinating question is whether one can identify the equations determining the 3-D complex surfaces of CP3× CP3 in turn determining space-time surfaces when one assigns to these surfaces M4× CP2 by mapping first these surfaces to the dual of CP3× CP3 and then applies sphere bundle projection to obtain the point of M4. Note that this formulation differs from the original one: I have been blundering a lot with the detailed realization of the idea.

1. The vanishing of three holomorphic functions fi would characterize 3-D holomorphic surfaces of 6-D CP3× CP3. These are determined by three real functions of three real arguments just like a holomorphic function of single variable is dictated by its values on a one-dimensional curve of complex plane. This conforms with the idea that initial data are given at 3-D surface.

2. Effective 2-dimensionality means that 2-D partonic surfaces plus 4-D tangent space data are enough. This suggests that the 2 holomorphic functions determining the dynamics satisfy some second order differential equation with respect to their three complex arguments: the value of the function and its derivative would correspond to the initial values of the imbedding space coordinates and their normal derivatives at partonic 2-surface. Since the effective 2-dimensionality brings in dependence on the induced metric of the space-time surface, this equation should contain information about the induced metric.

3. The no-where vanishing holomorphic 3-form Ω, which can be regarded as a "complex square root" of volume form characterizes 6-D Calabi-Yau manifold (see this), indeed contains this information albeit in a rather implicit manner but in spirit with TGD as almost topological QFT philosophy. Both CP3:s are characterized by this kind of 3-form if Calabi-Yau with (2,4) signature makes sense.

4. The simplest second order- and one might hope holomorphic- differential equation that one can imagine with these ingredients is of the form

Ω1i1j1k1×Ω2i2j2k2×∂i1i2 f1× ∂j1j2f2× ∂k1k2 f3=0 ,

ij== ∂ij .

Since Ωi is by its antisymmetry equal to Ωi123εijk, one can divide Ω123:s away from the equation so that one indeed obtains holomorphic solutions. Note also that one can replace ordinary derivatives in the equation with covariant derivatives without any effect so that the equations are general coordinate invariant.

5. The equations allow infinite families of obvious solutions. For instance, when some of i depends on the coordinates of either CP3 only, the equations are identically satisfied. As a special case one obtains solutions for which f2 depends on the coordinates of the first CP3 only and f3 only on those of the second CP3 and one has f1= Z • W=0. This solution family contains also the Calabi-Yau manifold found by Yau and Tian, whose factor space was proposed as the first candidate for a compactication consistent with three fermion families. The physical interpretation is of course completely different now.
Addition: I have now analyzed the general structure of the equations for the candidates for the lifts of the space-time surfaces. The equations when written for Taylor expansions are also bi-linear in the tensor product of linear spaces defined by Taylor coefficients for two complex coordinates variables so that one finds applications for the notion of hyper-determinant discussed in previous posting. The equations possess similar characteristic hierarchy as the deformations of vacuum extremals of Kähler action. This fact combined with the facts that Kähler action is nothing but Maxwell action for the induced Kähler form and that Penrose introduced twistors to describe solutions of Maxwell's equations give good hopes that the proposal might generalizes Penrose's work to non-linear context. Of course, the consistency with the number theoretic vision and with what is believed to be known about the general properties of preferred extremals poses extremely powerful constraints and it is difficult to avoid the feeling that a mathematical miracle is required.