Monday, September 20, 2010

M4×CP2 from pairs of twistors and twistorial formulation of TGD

In two previous postings (see this and this) I have considered the recent dramatic progress in twistor program from my humble vantage point defined by TGD. I am well aware of my technical limitations in these issuess but despite this I want to summarize some observations about the relationship between TGD and twistors which I should have discovered long time ago. There are also questions,ideas, and comments about the relationship to twistor string theory and M-theory and F-theory like structures. A more detailed discussion can be found at the end of the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework?. I apologize possible alert readers for the large amplitude critical fluctuations in the detailed formulation of the idea that the classical dynamics of TGD could allow description in terms of twistors.

The basic observations are following.

  1. Causal diamonds defined as intersections of future and past light-cones correspond to the Penrose diagrams lifted to representations of conformally compactified Minkowski space obtained by replacing the points of CD with spheres. The points of CD are representable by pairs of twistors and light-like points at boundaries of CD by single twistor.

  2. A pair of twistors defines a point defines a complex line of twistor space identifiable as a point of conformally compactified Minkowski space and thus of CD. Twistor itself is in turn mapped to CP2 in the dual twistor space CP3 by assigning to it the complex 2-plane defined by it via the linear equation Z•W=0 (in projective coordinates). Therefore the space CP3× CP3 is mapped naturally to M4×CP2 described in terms of the dual of CP3× CP3. It is however enough to use single pair CP3× CP3 if one wants to describe space-time surfaces as holomorphic surfaces. This suggests a deep relationship between the imbedding space of TGD and twistors which I have failed to realize hitherto!

  3. One can lift the partonic 2-surfaces at boundaries of CD to 4-D sphere bundles in twistor space CP3. This suggest that the twistor strings of Witten have a generalization in TGD framework to 6-D holomorphic surfaces in the product of twistor space and its dual. One can start from 12-D CP3 ×CP3, where the first CP3 represents projective twistors with projectively flat metric with signature (2,4) obtained from 8-D twistor space with signature (4,4). Second CP3 has Euclidian metric allowing Calabi-Yau structure. Canonically imbedded CP2 should have the standard metric with SU(3) actings as holonomies of the Calabi-Yau CP3 acting and as isometries of CP2.

  4. Also the first CP3 with (2,4) signature of conformal metric could allow a generalization of Calabi-Yau structure to Minkowskian signature (the Ricci tensor vanishes by conformal flatness) so that one would have something resembling F-theory with 2 time dimensions in the Cartesian product CP3 ×CP3 of twistors and dual twistors. The lifts of space-time surfaces in this space would be holomorphic 6-surfaces of this 12-D space. The consistency with M4 ×CP2 picture requires that these surfaces are sphere bundles with spheres projected to points of CD. Also the projections to the second CP3 must be consistent with the CP2 picture so that the holomorphic equations for complex plane are satisfied. The remaining two holomorphic equations would determine the dynamics of the space-time surface.

    The sphere bundle postulate reduces CP3×CP3 to CP3×CP2 and therefore leads to an analog of M-theory with two time directions. The further conditions imply that there is only one physical time-like direction. It must be emphasized that the super-symmetry in TGD framework is not same as in M-theory. For instance, the separate conservation of baryon and lepton numbers is a crucial distinction.

  5. Grassmannians have been suggested to have an identification in terms of the moduli spaces of twistor strings having representation as holomorphic surfaces. This conjecture should generalize appropriately so that partonic 2-surfaces with 4-D tangent space data or equivalently space-time surfaces (with certain restrictions) would bring in Grassmannians as part of the moduli spaces of their lifts to 6-D holomorphic surfaces in CP3×CP3.

Addition: Witten related the degree d of the algebraic curve describing twistor string, its genus g, the number k of negative helicity gluons, and the number l of loops by the formula

d=k-1+l g≤ l.

One should generalize the definition of the genus so that it applies to 6-D surfaces. For projective complex varieties of complex dimension n this definition indeed makes sense. Algebraic genus is expressible in terms of the dimensions of the spaces of closed holomorphic forms known as Hodge numbers hp,q as

g= ∑ (-1)n-khk,0 .

The first guess is that the formula of Witten generalizes by replacing genus with its algebraic counterpart. This requires that the allowed holomorphic surfaces are projective curves of twistor space, that is described in terms of homogenous polynomials of the 4+4 projective coordinates of CP3 ×CP3.

I do not bother to type further but give link to short file in which these observations are described in more detail. For an overall view about the proposed generalization of Yangian symmetry see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.


L. Edgar Otto said...


From what I think I understand of your presentation I see that your conclusions are correct on many points.

If what you mean by Grassmainian is the overlapping of such complex primes, that is correct but not enough.

I am much in favor of showing the links in convex spaces to these groups and explaining the deeper reasons for dimensional reductions, that each representation and how they work together is not just a matter of choice but tell us something significant.

I feel however that the deepest understanding must begin at 8 and not what some still think of as the vague idea of 6 dimensions. But for space as well as these abstract algebraic groups there must be more to it than we now imagine- we have most of the ingredients of a more general theory awaiting better interpretations.

Thank you for this series of posts.


Matti Pitkänen said...


Grassmannian G(n,k) is defined as the space of k-dimensional planes of n-dimensional space. It can be real or complex and the dimension is nk-1 (complex dimension in the latter case and in the recent situation). The duality G(n,k)= G(n,n-k) holds true.

For instance, for n=3 the space of lines in 3-D space is equal to G(3,1) equal to projective sphere which is sphere with opposite points identified. By duality this space is identical with the space of planes. Plane is indeed characterized by the line defining its normal.

In the case of scattering amplitudes one has complex Grassmannians having representations as coset spaces U(n)/U(n-k)xU(k). This representation and its analog in the real case follows by noticing that U(n) parametrizes coordinate complex frames and plane of given dimension is characterized by a pair of frames: frame for for plane and frame for normal space. The unitary rotations in normal space and plane however do not affect the plane so that the space of equivalence classes of these frame pairs parameterizes the plane and therefore one obtains the space in question.

For instance, CP_2 =U(3)/U2)xU(1) corresponds to the space of complex lines in 3-D complex space or equivalently the space of complex planes.

The origin of Grassmannians is not too well understood (at least by me!) but the natural idea is that they correspond to the moduli spaces of partonic 2-surfaces: this would generalize Witten's twistor string vision based on topological string theory: recall that TGD is almost topological QFT.

The deepest understanding indeed begins from D=8 as you say. My point is however that the formulation in terms of twistors might provide completely new insights as it indeed does. In particular, one can understand the uniqueness of M^4xCP_2 from twistorial point of view (besides number theoretically and from standard model symmetries): geometry, number theory and physics lead to to the same conclusion!

If the proposal for the description in terms of 6-D holomorphic makes sense it would be an enormous step forward in the understanding of classical TGD. In fact, it would be nothing less than an exact classical solution of field equations expressing as the conditions that these surfaces satisfy three holomorphic equations, two of which guarantee the M^4xCP_2 interpretation and the remaining ones fix the dynamics. This would generalize Penrose's original description of solutions of Maxwell's linear electrodynamics in terms of twistors to extremely non-linear situation.

This is probably too much to even dream about but crazy speculations are what I am paid for;-)!

Matti Pitkänen said...
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