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.

Basic idea

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.

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.

Basic idea

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 have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M⁴ and CP₂), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The signature of M⁴ metric is a problem. I have proposed complexification of M⁸ and M⁴ to get rid of the problem by assuming that the imbedding space corresponds to surfaces in the space M⁸ identified as octonions of form o₈= Re(o)+i Im(o), where o is imaginary part of ordinary octonion and i is commuting imaginary unit. M⁴ would correspond to quaternions of form q₄= Re(q)+i Im(q). What is important is that powers of q₄ and o₈ belong to this sub-space (as follows from the vanishing of cross product term in the square of octonion/quaternion) so that powers of q₄ (o₈) has imaginary part proportional to Im(q) (Im(o)).

I ended up to reconsider the idea of quaternion analyticity after having found 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 have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M⁴ and CP₂), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The first form of Cauchy-Rieman-Fueter conditions

Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.

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

   ∂_qf=(∂_t- ∂_x I-∂_y J -∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series. 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. The condition allows functions depending on complex coordinate z of some complex-plane only. It also allows functions satisfying two separate analyticity conditions, say
   

   ∂u*f= (∂_t- ∂_x I)f=0 ,
   

   ∂_v*f=-(∂_y J+∂_z K)f=-J(∂_y -∂ _zI) f=0 .
   

   In the latter formula J multiplies from left! One has good hopes of obtaining holomorphic functions of two complex coordinates. This might be enough to understand the preferred extremals of Kähler action as quaternion analytic mops.
   

   
   There are potential problems due to non-commutativity of u=t+/- xI and v=yJ+/- zK= (y+/- zI) J (note that J ~ multiplies from right!) and ∂_u and ∂_v. A prescription for the ordering of the powers u and v in the polynomials of u and v appearing in the double Taylor series seems to be needed. For instance, powers of u can be taken to be at left and v or of a related variable at right.
   

   By the linearity of ∂_v* one can leave J to the left and commute only (∂_y -∂ _zI) through the u-dependent part of the series: this operation is trivial. The condition ∂_vf=0 is satisfied if the polynomials of y and z are polynomials of y+iz multiplied by J from right. The solution ansatz is thus product of Taylor series of monomials f_mn= (x+iy)^m
   (y+iz)ⁿJ with Taylor coefficients a_mn, which multiply the monomials from right
   and are arbitrary quaternions. Note that the monomials (y+iz)ⁿ do not reduce to polynomials of v and that the ordering of these powers is arbitrary. If the coefficients a_mn are real f maps 4-D quaternionic region to 2-D region spanned by J and K. Otherwise the image is 4-D.
   
   

3. By linearity the solutions obey linear superposition. They can be also multiplied if product is defined as ordered product in such a manner that only the powers t+ix and y+iz are multiplied together at left and coefficients a_mn are multiplied together at right. The analogy with quantum non-commutativity is obvious.
   

   
   

4. In Minkowskian signature one must multiply imaginary units I,J,K with an additional commuting imaginary unit i. This would give solutions as powers of (say) t+ex, e=iI with e²=1 representing imaginary unit of hyper-complex numbers. The natural interpretation would be as algebraic extension which is analogous to the extension of rational number by adding algebraic number, say 2^1/2 to get algebraically 2-dimensional structure but as real numbers 1-D structure. Only the non-commutativity with J and K distinguishes e from e=+/- 1 and if J and K do not appear in the function, one can replace e by +/- 1 in t+ex to get just t+/- x appearing as argument for waves propagating with light velocity.
   
   

Second form of CRF conditions

Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.

1. The ingenious idea is to replace preferred quaternionic imaginary unit by a imaginary unit which is in radial direction: e_r= (xI+yJ+zK)/r and require analyticity with respect to the coordinate t+e r. The solution to the condition is power series in t+re_r= q so that one obtains quaternion analyticity.
   
   
2. The excplicit form of the conditions is
   

   l (∂_t- e_r ∂_r)f= (∂_t-e_r/r r∂_r)f=0 .
   

   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.
   
   

3. 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.
   
   
4. 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 CD 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 correspond to the radial Eguchi-Hanson coordinate of CP₂ and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

5. Also in this case one can ask whether solutions depending on two complex local coordinates analogous to those for translationally invariant CRF condition are possible. The remain imaginary units would be associated with the surface of sphere allowing complex structure.
   
   

Generalization of CRF conditions?

Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?

1. Ordinary complex analysis suggests that there is an infinite number of choices of the quaternionic coordinates related by the above described quaternion-analytic maps with 4-D images. The form of of the CRF conditions would be different in each of these coordinate systems and would be obtained in a straightforward manner by chain rule.
   
   
2. One expects the existence of large number of different quaternion-conformal structures not related by quaternion-analytic transformations analogous to those allowed by higher genus Riemann surfaces and that these conformal equivalence classes of four-manifolds are characterized by a moduli space and the analogs of Teichmueller
   parameters depending on 3-topology. In TGD framework strong form of holography suggests that these conformal equivalence classes for preferred extremals could reduce to ordinary conformal classes for the partonic 2-surfaces. An attractive possibility is that by conformal gauge symmetries the functional integral over WCW reduces to the integral over the conformal equivalence classes.
   
   
3. The quaternion-conformal structures could be characterized by a standard choice of quaternionic coordinates reducing to the choice of a pair of complex coordinates. In these coordinates the general solution to quaternion-analyticity conditions would be of form described for the linear ansatz. The moduli space corresponds to that for complex or hyper-complex structures defined in the space-time region.
   
   

Geometric formulation of the CRF conditions

The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.

In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.

The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors J_u and J_u representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as

J- ∑_u J_u J j_u=0 .

For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λ^uv to obtain

J- Λ^uv J_uJ j_v=0 .

The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.

Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.

In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.

Does residue calculus generalize?

CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.

Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.

Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.

Could one understand the preferred extremals in terms of quaternion-analyticity?

Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X⁴ to quaternionic CP₂ and from quaternionic X⁴ to hyper-quaternionic M⁴? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.

The best thing to do is to look whether this is possible for the known extremals: CP₂ type vacuum extremals, vacuum extremals expressible as graph of map from M⁴ to a Lagrangian sub-manifold of CP₂, cosmic strings of form X²× Y²⊂ M⁴ × CP₂ such that X² is string world sheet (minimal surface) and Y² complex sub-manifold of CP₂. One can also check whether Hamilton-Jacobi structure of M⁴ and of Minkowskian space-time regions and complex structure of CP₂ have natural counterparts in the quaternion-analytic framework.

1. Consider first cosmic strings. In this case the quaternionic-analytic map from X⁴ = X²× Y² to M⁴× CP₂ with octonion structure would be map X⁴ to 2-D string world sheet in M² and Y² to 2-D complex manifold of CP₂. This could be achieved by using the linear variant of CRF condition. The map from X⁴ to M⁴ would reduce to ordinary hyper-analytic map from X² with hyper-complex coordinate to M⁴ with hyper-complex coordinates just as in string models. The map from X⁴ to CP₂ would reduce to an ordinary analytic map from X² with complex coordinates. One would not leave the realm of string models.
   
   
2. For the simplest massless extremals (MEs) CP₂ coordinates are arbitrary functions of light-like coordinate u=k•m, k constant light-like vector, and of v=ε • m, ε a polarization vector orthogonal to k. The interpretation as classical counterpart of photon or Bose-Einstein condense of photons is obvious. There are good reasons to expect that this ansatz generalizes by replacing the variables u and v with coordinate along the light-like and space-like coordinate lines of Hamilton-Jacobi structure. The non-geodesic motion of photons with light-velocity and variation of the polarization direction would be due to interactions with the space-time sheet to which it is topologically condensed. Note that light-likeness condition for the coordinate curve gives rise to Virasoro conditions. This observation led long time ago to the idea that 2-D conformal invariance must have a non-trivial generalization to 4-D case.
   

   Now space-time surface would have naturally M⁴ coordinates and the map M⁴→ M⁴ would be just identity map satisfying the radial CRF condition. Can one understand CP₂ coordinates in terms of quaternion- analyticity? The dependence of CP₂ coordinates on u=t-x only can be formulated as CFR condition ∂_u* s^k=0 and this could is expected to generalize in the formulation using the geometric representation of quaternionic imaginary units at both sides. The dependence on light-light coordinate u follows from the translationally invariant CRF condition.
   

   The dependence on the real coordinate v is however problematic since the dependence is naturally on complex coordinate w assignable to the polarization plane of form z= f(w). This would give dependence on 2 transversal coordinates and CP₂ projection would be 3-D rather than 2-D. One can of course ask whether this dependence is actually present for preferred extremals? Could the polarization vector be complex local polarization vector orthogonal to the light-like vector? In quantum theory complex polarization vectors are used of routinely and become oscillator operators in second quantization and in TGD Universe MEs indeed serve as space-time correlates for photons or their BE condensates.
   
   

3. Vacuum extremals with Lagrangian manifold as (in the generic case 2-D) CP₂ projection are not expected to be preferred extremals for obvious reasons. One one can however try similar approach. Hyper-quaternionic structure for space-time surface using Hamilton-Jacobi structure is the first guess. CP₂ should allow a quaternionic coordinate decomposing to a pair of complex coordinates such that second complex coordinate is constant for 2-D Lagrangian manifold and second parameterizes it. Any 2-D surface allows complex structure defined by the induced metric so that there are good hopes that these coordinates exist. The quaternion-analytic map would map in the most general case is trivial for both hypercomplex and complex coordinate of M⁴ but the quaternionic Taylor coefficients reduce to real numbers to that the image is 2-D.
   
   
4. For CP₂ type vacuum extremals the M⁴ projection is random light-like curve. Now one expects co-quaternionicity and that quaternion-analyticity is not the correct manner to formulate the situation. "Co-" suggests that instead of expressing surface as graph one should perhaps express it in terms of conditions stating that some quaternionic analytic functions in H are vanish.
   

   One can fix the coordinates of X⁴ to be complex coordinates of CP₂ so that one gets rid of the degeneracy due to the choice of coordinates. M⁴ allows hyper-quaternionic coordinates and Hamilton-Jacobi structures define different choices of hyper-quaternionic coordinates. Now the second light- like coordinate would vary along random light-like curves providing slicing of M⁴ by 3-D surfaces.
   Hamilton-Jacobi structure defines at each point a plane M²(x) fixed by the light-like vector at the point and the 2-D orthogonal plane. In fact 4-D coordinate grid is defined. This local choice must be integrable, which means that one has slicing by 2-D string world sheets and polarization planes orthogonal to them.
   

   The problem is that the mapping of quaternionic CP₂ coordinate to hyper-quaternionic coordinates of M⁴ (say v=0, w=0) in terms of quaternionic analyticity is not easy. "Co-" suggets that , one could formulate light-likeness condition using Hamilton-Jacobi structure as conditions w*-constant=0 and v-constant=0. Note that one has u*=v.
   
   

5. In the naive generalization CRF conditions are linear. Whether this is the case in the formulation using the geometric representation of the imaginary units is not clear since the quaternionic imaginary units depend on the vielbein of the induced 3-metric (note however that the SO(3) gauge rotation appearing in the conditions could allow to compensate for the change of the tensors in small deformations of the spaced-time surface). If linearity is real and not true only for small perturbations, one could have linear superpositions of different types of solutions, which looks strange. Could the superpositions describe perturbations of say cosmic strings and massless extremals?
   
   
6. Both forms of algebraic 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 gauge invariance to 4-D context?
   

Conclusions

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.

The first form of Cauchy-Rieman-Fueter conditions

Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.

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

   ∂_qf=(∂_t- ∂_x I-∂_y J -∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series. 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. The condition allows functions depending on complex coordinate z of some complex-plane only. It also allows functions satisfying two separate analyticity conditions, say
   

   ∂u*f= (∂_t- ∂_x I)f=0 ,
   

   ∂_v*f=-(∂_y J+∂_z K)f=-J(∂_y -∂ _zI) f=0 .
   

   In the latter formula J multiplies from left! One has good hopes of obtaining holomorphic functions of two complex coordinates. This might be enough to understand the preferred extremals of Kähler action as quaternion analytic mops.
   

   
   There are potential problems due to non-commutativity of u=t+/- xI and v=yJ+/- zK= (y+/- zI) J (note that J ~ multiplies from right!) and ∂_u and ∂_v. A prescription for the ordering of the powers u and v in the polynomials of u and v appearing in the double Taylor series seems to be needed. For instance, powers of u can be taken to be at left and v or of a related variable at right.
   

   By the linearity of ∂_v* one can leave J to the left and commute only (∂_y -∂ _zI) through the u-dependent part of the series: this operation is trivial. The condition ∂_vf=0 is satisfied if the polynomials of y and z are polynomials of y+iz multiplied by J from right. The solution ansatz is thus product of Taylor series of monomials f_mn= (x+iy)^m
   (y+iz)ⁿJ with Taylor coefficients a_mn, which multiply the monomials from right
   and are arbitrary quaternions. Note that the monomials (y+iz)ⁿ do not reduce to polynomials of v and that the ordering of these powers is arbitrary. If the coefficients a_mn are real f maps 4-D quaternionic region to 2-D region spanned by J and K. Otherwise the image is 4-D.
   
   

3. By linearity the solutions obey linear superposition. They can be also multiplied if product is defined as ordered product in such a manner that only the powers t+ix and y+iz are multiplied together at left and coefficients a_mn are multiplied together at right. The analogy with quantum non-commutativity is obvious.
   

   
   

4. In Minkowskian signature one must multiply imaginary units I,J,K with an additional commuting imaginary unit i. This would give solutions as powers of (say) t+ex, e=iI with e²=1 representing imaginary unit of hyper-complex numbers. The natural interpretation would be as algebraic extension which is analogous to the extension of rational number by adding algebraic number, say 2^1/2 to get algebraically 2-dimensional structure but as real numbers 1-D structure. Only the non-commutativity with J and K distinguishes e from e=+/- 1 and if J and K do not appear in the function, one can replace e by +/- 1 in t+ex to get just t+/- x appearing as argument for waves propagating with light velocity.
   
   

Second form of CRF conditions

Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.

1. The ingenious idea is to replace preferred quaternionic imaginary unit by a imaginary unit which is in radial direction: e_r= (xI+yJ+zK)/r and require analyticity with respect to the coordinate t+e r. The solution to the condition is power series in t+re_r= q so that one obtains quaternion analyticity.
   
   
2. The excplicit form of the conditions is
   

   l (∂_t- e_r ∂_r)f= (∂_t-e_r/r r∂_r)f=0 .
   

   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.
   
   

3. 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.
   
   
4. 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 CD 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 correspond to the radial Eguchi-Hanson coordinate of CP₂ and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

5. Also in this case one can ask whether solutions depending on two complex local coordinates analogous to those for translationally invariant CRF condition are possible. The remain imaginary units would be associated with the surface of sphere allowing complex structure.
   
   

Generalization of CRF conditions?

Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?

1. Ordinary complex analysis suggests that there is an infinite number of choices of the quaternionic coordinates related by the above described quaternion-analytic maps with 4-D images. The form of of the CRF conditions would be different in each of these coordinate systems and would be obtained in a straightforward manner by chain rule.
   
   
2. One expects the existence of large number of different quaternion-conformal structures not related by quaternion-analytic transformations analogous to those allowed by higher genus Riemann surfaces and that these conformal equivalence classes of four-manifolds are characterized by a moduli space and the analogs of Teichmueller
   parameters depending on 3-topology. In TGD framework strong form of holography suggests that these conformal equivalence classes for preferred extremals could reduce to ordinary conformal classes for the partonic 2-surfaces. An attractive possibility is that by conformal gauge symmetries the functional integral over WCW reduces to the integral over the conformal equivalence classes.
   
   
3. The quaternion-conformal structures could be characterized by a standard choice of quaternionic coordinates reducing to the choice of a pair of complex coordinates. In these coordinates the general solution to quaternion-analyticity conditions would be of form described for the linear ansatz. The moduli space corresponds to that for complex or hyper-complex structures defined in the space-time region.
   
   

Geometric formulation of the CRF conditions

The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.

In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.

The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors J_u and J_u representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as

J- ∑_u J_u J j_u=0 .

For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λ^uv to obtain

J- Λ^uv J_uJ j_v=0 .

The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.

Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.

In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.

Does residue calculus generalize?

CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.

Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.

Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.

Could one understand the preferred extremals in terms of quaternion-analyticity?

Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X⁴ to quaternionic CP₂ and from quaternionic X⁴ to hyper-quaternionic M⁴? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.

The best thing to do is to look whether this is possible for the known extremals: CP₂ type vacuum extremals, vacuum extremals expressible as graph of map from M⁴ to a Lagrangian sub-manifold of CP₂, cosmic strings of form X²× Y²⊂ M⁴ × CP₂ such that X² is string world sheet (minimal surface) and Y² complex sub-manifold of CP₂. One can also check whether Hamilton-Jacobi structure of M⁴ and of Minkowskian space-time regions and complex structure of CP₂ have natural counterparts in the quaternion-analytic framework.

1. Consider first cosmic strings. In this case the quaternionic-analytic map from X⁴ = X²× Y² to M⁴× CP₂ with octonion structure would be map X⁴ to 2-D string world sheet in M² and Y² to 2-D complex manifold of CP₂. This could be achieved by using the linear variant of CRF condition. The map from X⁴ to M⁴ would reduce to ordinary hyper-analytic map from X² with hyper-complex coordinate to M⁴ with hyper-complex coordinates just as in string models. The map from X⁴ to CP₂ would reduce to an ordinary analytic map from X² with complex coordinates. One would not leave the realm of string models.
   
   
2. For the simplest massless extremals (MEs) CP₂ coordinates are arbitrary functions of light-like coordinate u=k•m, k constant light-like vector, and of v=ε • m, ε a polarization vector orthogonal to k. The interpretation as classical counterpart of photon or Bose-Einstein condense of photons is obvious. There are good reasons to expect that this ansatz generalizes by replacing the variables u and v with coordinate along the light-like and space-like coordinate lines of Hamilton-Jacobi structure. The non-geodesic motion of photons with light-velocity and variation of the polarization direction would be due to interactions with the space-time sheet to which it is topologically condensed. Note that light-likeness condition for the coordinate curve gives rise to Virasoro conditions. This observation led long time ago to the idea that 2-D conformal invariance must have a non-trivial generalization to 4-D case.
   

   Now space-time surface would have naturally M⁴ coordinates and the map M⁴→ M⁴ would be just identity map satisfying the radial CRF condition. Can one understand CP₂ coordinates in terms of quaternion- analyticity? The dependence of CP₂ coordinates on u=t-x only can be formulated as CFR condition ∂_u* s^k=0 and this could is expected to generalize in the formulation using the geometric representation of quaternionic imaginary units at both sides. The dependence on light-light coordinate u follows from the translationally invariant CRF condition.
   

   The dependence on the real coordinate v is however problematic since the dependence is naturally on complex coordinate w assignable to the polarization plane of form z= f(w). This would give dependence on 2 transversal coordinates and CP₂ projection would be 3-D rather than 2-D. One can of course ask whether this dependence is actually present for preferred extremals? Could the polarization vector be complex local polarization vector orthogonal to the light-like vector? In quantum theory complex polarization vectors are used of routinely and become oscillator operators in second quantization and in TGD Universe MEs indeed serve as space-time correlates for photons or their BE condensates.
   
   

3. Vacuum extremals with Lagrangian manifold as (in the generic case 2-D) CP₂ projection are not expected to be preferred extremals for obvious reasons. One one can however try similar approach. Hyper-quaternionic structure for space-time surface using Hamilton-Jacobi structure is the first guess. CP₂ should allow a quaternionic coordinate decomposing to a pair of complex coordinates such that second complex coordinate is constant for 2-D Lagrangian manifold and second parameterizes it. Any 2-D surface allows complex structure defined by the induced metric so that there are good hopes that these coordinates exist. The quaternion-analytic map would map in the most general case is trivial for both hypercomplex and complex coordinate of M⁴ but the quaternionic Taylor coefficients reduce to real numbers to that the image is 2-D.
   
   
4. For CP₂ type vacuum extremals the M⁴ projection is random light-like curve. Now one expects co-quaternionicity and that quaternion-analyticity is not the correct manner to formulate the situation. "Co-" suggests that instead of expressing surface as graph one should perhaps express it in terms of conditions stating that some quaternionic analytic functions in H are vanish.
   

   One can fix the coordinates of X⁴ to be complex coordinates of CP₂ so that one gets rid of the degeneracy due to the choice of coordinates. M⁴ allows hyper-quaternionic coordinates and Hamilton-Jacobi structures define different choices of hyper-quaternionic coordinates. Now the second light- like coordinate would vary along random light-like curves providing slicing of M⁴ by 3-D surfaces.
   Hamilton-Jacobi structure defines at each point a plane M²(x) fixed by the light-like vector at the point and the 2-D orthogonal plane. In fact 4-D coordinate grid is defined. This local choice must be integrable, which means that one has slicing by 2-D string world sheets and polarization planes orthogonal to them.
   

   The problem is that the mapping of quaternionic CP₂ coordinate to hyper-quaternionic coordinates of M⁴ (say v=0, w=0) in terms of quaternionic analyticity is not easy. "Co-" suggets that , one could formulate light-likeness condition using Hamilton-Jacobi structure as conditions w*-constant=0 and v-constant=0. Note that one has u*=v.
   
   

5. In the naive generalization CRF conditions are linear. Whether this is the case in the formulation using the geometric representation of the imaginary units is not clear since the quaternionic imaginary units depend on the vielbein of the induced 3-metric (note however that the SO(3) gauge rotation appearing in the conditions could allow to compensate for the change of the tensors in small deformations of the spaced-time surface). If linearity is real and not true only for small perturbations, one could have linear superpositions of different types of solutions, which looks strange. Could the superpositions describe perturbations of say cosmic strings and massless extremals?
   
   
6. Both forms of algebraic 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 gauge invariance to 4-D context?
   

Conclusions

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.