In blog comments Anonymous gave a link to an article about construction of 4D quaternionKähler metrics with an isometry: they are determined by so called SU(∞) Toda equation. I tried to see whether quaternionKähler 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 2ndimensional space (now real). In 4D case tangent space contains 3D submanifold identifiable as imaginary quaternions. CP_{2} is one example of QK manifold for which the subgroup in question is SU(2)× U(1) and which has nonvanishing 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 (6D) 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?
 In the twistor formulation of TGD the spacetime surfaces are 4D base spaces of 6D twistor spaces in the Cartesian product of 6D twistor spaces of M^{4} and CP_{2}  the only twistor spaces with Kähler structure. In TGD framework spacetime regions can have either Euclidian or Minkowskian signature of induced metric. The lines of generalized Feynman diagrams have Euclidian signature.
 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 HamiltonJacobi 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.
 Why it would be natural to have QK that is corresponding twistor space which is projective contact Fano manifold?
 Fano property implies that the 4D Euclidian spacetime region representing line of the Feynman diagram
can be imbedded as a submanifold 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 spacetime correlate for the fact that CP_{n}s emerge in twistor Grassmann approach expected to generalize to TGD framework.
 CP_{2} allows both projective (trivially) and contact (even symplectic) structures. δ M^{4}_{+} × CP_{2} allows contact structure  I call it loosely symplectic structure. Also 3D lightlike orbits of partonic 2surfaces allow contact structure. Therefore holomorphic contact structure for the twistor space is natural.
 Both the holomorphic contact structure and projectivity of CP_{2} would be inherited if QK property is true. Contact structures at orbits of partonic 2surfaces would extend to holomorphic contact structures in the Euclidian regions of spacetime surface representing lines of generalized Feynman diagrams. Projectivity of Fano space would be also inherited from CP_{2} 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).
 Could the isometry (or possibly isometries) for QK be seen as a remnant of color symmetry or rotational symmetries of M^{4} 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 CP_{2} center of mass degrees of freedom). Could the isometry correspond to anomalous hypercharge?
How to choose the quaternionic imaginary units for the spacetime surface?
Parallellizability is a very special property of 3manifolds allowing to choose quaternionic imaginary units: global choice of one of them gives rise to twistor structure.
 The selection of time coordinate defines a slicing of spacetime surface by 3surfaces. GCI would suggest that a generic slicing gives rise to 3 quaternionic units at each point each 3surface? The parallelizability of 3manifolds  a unique property of 3manifolds  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 tribein or its dual defined by twoforms obtained by contracting tribein vectors with permutation tensor gives the quanternionic imaginary units. The construction depends on 3metric only and could be carried out also in GRT context. Note however that topology change for 3manifold might cause some nontrivialities. The metric 2dimensionality at the lightlike orbits of partonic 2surfaces should not be a problem for a slicing by spacelike 3surfaces. The construction makes sense also for the regions of Minkowskian signature.
 In zero energy ontology (ZEO) a purely TGD based feature  there are very natural special slicings. The first one is by linear timelike Minkowski coordinate defined by the direction of the line connecting the tips of the causal diamond (CD). Second one is defined by the lightcone proper time associated with either lightcone in the intersection of future and past directed lightcones defining CD. Neither slicing is global as it is easy to see.
The relationship to quaternionicity conjecture and M^{8}H duality
One of the basic conjectures of TGD is that preferred extremals consist of quaternionic/ coquaternionic (associative/coassociative) regions (see this). Second closely related conjecture is M^{8}H duality allowing to map quaternionic/coquaternionic surfaces of M^{8} to those of M^{4}× CP_{2}. Are these
conjectures consistent with QK in Euclidian regions and HamiltonJacobi property in Minkowskian regions? Consider first the definition of quaternionic and coquaternionic spacetime regions.
 Quaternionic/associative spacetime region (with Minkowskian signature) is defined in terms of induced octonion structure obtained by projecting octonion units defined by vielbein of H= M^{4}× CP_{2} to spacetime surface and demanding that the 4 projections generate quaternionic subalgebra at each point of spacetime.
If there is also unique complex subalgebra associated with each point of spacetime, one obtains one can assign to the tangent spaceof spacetime surface a point of CP_{2}. This allows to realize M^{8}H duality (see this) as the number theoretic analog of spontaneous compactification (but involving no compactification) by assigning to a point of M^{4}=M^{4}× CP_{2} a point of M^{4}× CP_{2}. If the image surface is also quaternionic, this assignment makes sense also for spacetime surfaces in H so that M^{8}H duality generalizes to HH 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.
 Coquaternionic/coassociative structure is conjectured for spacetime regions of Euclidian signature and 4D CP_{2} projection. In this case normal space of spacetime 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.
 Quaternionicity in this sense requires unique identification of a preferred time coordinate as imbedding space coordinate and corresponding slicing by 3surfaces 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.
 The basis of induced octonionic imaginary unit allows to identify quaternionic imaginary units linearly related to the corresponding units defined by tribein 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.
 The selection of the preferred imaginary quaternion unit is necessary for M^{8}H
correspondence. This selection would also define the twistor structure. For quaternionKähler manifold this unit would be covariantly constant and define Kähler form  maybe as the induced Kähler form.
 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 HamiltonJacobi structure consistent with this?
HamiltonJacobi structure involves a selection of 2D complex plane at each point of spacetime surface. Could induced Kähler magnetic form for each 3slice define this plane? It is not necessary to require that 3D 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 timedependent Kählerelectric 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?
See the chapter Classical part of the twistor story or the article Classical part of the twistor story.
