Could the lines of generalised Feynman diagrams correspond to quaternion-Kähler manifolds?

In blog comments Anonymous gave a link to an article about construction of 4-D quaternion-Kähler metrics with an isometry: they are determined by so called SU(∞) Toda equation. I tried to see whether quaternion-Kähler manifolds could be relevant for TGD.

From Wikipedia one can learn that QK is characterized by its holonomy, which is a subgroup of Sp(n)×Sp(1): Sp(n) acts as linear symplectic transformations of 2n-dimensional space (now real). In 4-D case tangent space contains 3-D sub-manifold identifiable as imaginary quaternions. CP₂ is one example of QK manifold for which the subgroup in question is SU(2)× U(1) and which has non-vanishing constant curvature: the components of Weyl tensor represent the quaternionic imaginary units. QKs are Einstein manifolds: Einstein tensor is proportional to metric.

What is really interesting from TGD point of view is that twistorial considerations show that one can assign to QK a special kind of twistor space (twistor space in the mildest sense requires only orientability). Wiki tells that if Ricci curvature is positive, this (6-D) twistor space is what is known as projective Fano manifold with a holomorphic contact structure. Fano variety has the nice property that as (complex) line bundle it has enough sections to define the imbedding of its base space to a projective variety. Fano variety is also complete: this is algebraic geometric analogy of topological property known as compactness.

QK manifolds and twistorial formulation of TGD

How the QKs could relate to the twistorial formulation of TGD?

1. In the twistor formulation of TGD the space-time surfaces are 4-D base spaces of 6-D twistor spaces in the Cartesian product of 6-D twistor spaces of M⁴ and CP₂ - the only twistor spaces with Kähler structure. In TGD framework space-time regions can have either Euclidian or Minkowskian signature of induced metric. The lines of generalized Feynman diagrams have Euclidian signature.
   
   
2. Could the twistor spaces associated with the lines of generalized Feynman diagrams be projective Fano manifolds? Could QK structure characterize Euclidian regions of preferred extremals of Kähler action. Could a generalization to Minkowskian regions exist. I have proposed that so called Hamilton-Jacobi structure characterizes preferred extremals in Minkowskian regions. It could be the natural Minkowskian counterpart for the quaternion Kähler structure, which involves only imaginary quaternions and could make sense also in Minkowski signature. Note that unit sphere of imaginary quaternions defines the sphere serving as fiber of the twistor bundle.
   
   
3. Why it would be natural to have QK that is corresponding twistor space which is projective contact Fano manifold?
   
   
   1. Fano property implies that the 4-D Euclidian space-time region representing line of the Feynman diagram can be imbedded as a sub-manifold to complex projective space CP_n. This would allow to use the powerful machinery of projective geometry in TGD framework. This could also be a space-time correlate for the fact that CP_ns emerge in twistor Grassmann approach expected to generalize to TGD framework.
      
      
   2. CP₂ allows both projective (trivially) and contact (even symplectic) structures. δ M⁴₊ × CP₂ allows contact structure - I call it loosely symplectic structure. Also 3-D light-like orbits of partonic 2-surfaces allow contact structure. Therefore holomorphic contact structure for the twistor space is natural.
      
      
   3. Both the holomorphic contact structure and projectivity of CP₂ would be inherited if QK property is true. Contact structures at orbits of partonic 2-surfaces would extend to holomorphic contact structures in the Euclidian regions of space-time surface representing lines of generalized Feynman diagrams. Projectivity of Fano space would be also inherited from CP₂ or its twistor space SU(3)/U(1)× U(1) (flag manifold identifiable as the space of choices for quantization axes of color isospin and hypercharge).
      
      
4. Could the isometry (or possibly isometries) for QK be seen as a remnant of color symmetry or rotational symmetries of M⁴ factor of imbedding space? The only remnant of color symmetry at the level of imbedding space spinors is anomalous color hyper charge (color is like orbital angular momentum and associated with spinor harmonic in CP₂ center of mass degrees of freedom). Could the isometry correspond to anomalous hypercharge?
   
   

How to choose the quaternionic imaginary units for the space-time surface?

Parallellizability is a very special property of 3-manifolds allowing to choose quaternionic imaginary units: global choice of one of them gives rise to twistor structure.

1. The selection of time coordinate defines a slicing of space-time surface by 3-surfaces. GCI would suggest that a generic slicing gives rise to 3 quaternionic units at each point each 3-surface? The parallelizability of 3-manifolds - a unique property of 3-manifolds - means the possibility to select global coordinate frame as section of the frame bundle: one has 3 sections of tangent bundle whose inner products give rose to the components of the metric (now induced metric) guarantees this. The tri-bein or its dual defined by two-forms obtained by contracting tri-bein vectors with permutation tensor gives the quanternionic imaginary units. The construction depends on 3-metric only and could be carried out also in GRT context. Note however that topology change for 3-manifold might cause some non-trivialities. The metric 2-dimensionality at the light-like orbits of partonic 2-surfaces should not be a problem for a slicing by space-like 3-surfaces. The construction makes sense also for the regions of Minkowskian signature.
   
   
2. In zero energy ontology (ZEO)- a purely TGD based feature - there are very natural special slicings. The first one is by linear time-like Minkowski coordinate defined by the direction of the line connecting the tips of the causal diamond (CD). Second one is defined by the light-cone proper time associated with either light-cone in the intersection of future and past directed light-cones defining CD. Neither slicing is global as it is easy to see.
   
   

The relationship to quaternionicity conjecture and M⁸-H duality

One of the basic conjectures of TGD is that preferred extremals consist of quaternionic/ co-quaternionic (associative/co-associative) regions (see this). Second closely related conjecture is M⁸-H duality allowing to map quaternionic/co-quaternionic surfaces of M⁸ to those of M⁴× CP₂. Are these conjectures consistent with QK in Euclidian regions and Hamilton-Jacobi property in Minkowskian regions? Consider first the definition of quaternionic and co-quaternionic space-time regions.

1. Quaternionic/associative space-time region (with Minkowskian signature) is defined in terms of induced octonion structure obtained by projecting octonion units defined by vielbein of H= M⁴× CP₂ to space-time surface and demanding that the 4 projections generate quaternionic sub-algebra at each point of space-time.
   

   If there is also unique complex sub-algebra associated with each point of space-time, one obtains one can assign to the tangent space-of space-time surface a point of CP₂. This allows to realize M⁸-H duality (see this) as the number theoretic analog of spontaneous compactification (but involving no compactification) by assigning to a point of M⁴=M⁴× CP₂ a point of M⁴× CP₂. If the image surface is also quaternionic, this assignment makes sense also for space-time surfaces in H so that M⁸-H duality generalizes to H-H duality allowing to assign to given preferred extremal a hierarchy of extremals by iterating this assignment. One obtains a category with morphisms identifiable as these duality maps.
   
   

2. Co-quaternionic/co-associative structure is conjectured for space-time regions of Euclidian signature and 4-D CP₂ projection. In this case normal space of space-time surface is quaternionic/associative. A multiplication of the basis by preferred unit of basis gives rise to a quaternionic tangent space basis so that one can speak of quaternionic structure also in this case.
   
   
3. Quaternionicity in this sense requires unique identification of a preferred time coordinate as imbedding space coordinate and corresponding slicing by 3-surfaces and is possible only in TGD context. The preferred time direction would correspond to real quaternionic unit. Preferred time coordinate implies that quaternionic structure in TGD sense is more specific than the QK structure in Euclidian regions.
   
   
4. The basis of induced octonionic imaginary unit allows to identify quaternionic imaginary units linearly related to the corresponding units defined by tri-bein vectors. Note that the multiplication of octonionic units is replaced with multiplication of antisymetric tensors representing them when one assigns to the quaternionic structure potential QK structure. Quaternionic structure does not require Kähler structure and makes sense for both signatures of the induced metric. Hence a consistency with QK and its possible analog in Minkowskian regions is possible.
   
   
5. The selection of the preferred imaginary quaternion unit is necessary for M⁸-H correspondence. This selection would also define the twistor structure. For quaternion-Kähler manifold this unit would be covariantly constant and define Kähler form - maybe as the induced Kähler form.
   
   
6. Also in Minkowskian regions twistor structure requires a selection of a preferred imaginary quaternion unit. Could the induced Kähler form define the preferred imaginary unit also now? Is the Hamilton-Jacobi structure consistent with this?
   

   Hamilton-Jacobi structure involves a selection of 2-D complex plane at each point of space-time surface. Could induced Kähler magnetic form for each 3-slice define this plane? It is not necessary to require that 3-D Kähler form is covariantly constant for Minkowskian regions. Indeed, massless extremals representing analogs of photons are characterized by local polarization and momentum direction and carry time-dependent Kähler-electric and -magnetic fields. One can however ask whether monopole flux tubes carry covariantly constant Kähler magnetic field: they are indeed deformations of what I call cosmic strings (see this) for which this condition holds true?
   

Could quaternion analyticity make sense for the preferred extremals?

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.

One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M⁸ or E⁸ coordinate, which is enough if M⁸-H duality holds true) would for preferred extremals be expressible in the form

o(q)= u(q) + v(q)× I .

Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.

I ended up to this idea after finding two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I managed to miss Fueter's work), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The so called Cauhy-Riemann-Fueter conditions generalize Cauchy-Riemann conditions. These conditions are however not unique.

1. The translationally invariant form of CRF conditions is ∂_q*
   f=0 or explicitly
   

   (∂_o+ ∂_x I+∂_yJ y+∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series although it allows functions depending on complex coordinate z of some complex-plane only.
   

   Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.
   

2. Second form of CRF conditions is
   

   (∂₀+ (xI+yJ+zK)/r²) r∂_r)f=0 .
   

   Here r denotes the imaginary part of q. This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.
   

   This form of CRF is neither Lorentz invariant nor translationally invariant but remains
   invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.
   
   

3. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M⁴_+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M⁴_+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CDa defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.
   

   In regions of Euclidian signature r could corresponds to the the radial Eguchi-Hanson coordinates and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

4. In TGD framework one must have also ordinary holomorphic functions mapping 4-D quaterionic space to 2-D complex space associated with real and/or imaginary part of octonion: they would be associated with extremals for which M⁴ and/or CP₂ projection is 2-dimensional (cosmic strings and massless extremals).
   

   Since the conditions are linear one can also have superpositions of different types of solutions and they could describe perturbations of say cosmic strings and massless extremals.
   
   

5. It is of course possible that an entire moduli space of tri-holomorphic operators exists and would be interesting to know the most general form of tri-holomorphic operator. For instance, if one performs a quaternion analytic map of the space-time surface the form of the operator defining C-R-F conditions changes and becomes rather complex. This suggests that the operator as it is defined above indeed refers to geometrically preferred coordinates as already suggested. One must be however very cautious. It might well be that quaternion conformal transforms of this operator are possible but that they are equivalent.
   
   
6. Both generalizations of the C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.
   

   Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal invariance to 4-D context?
   

To sum up, connections between different conjectures related to the preferred extremals - M⁸-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.