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M^{8}-H duality, preferred extremals, criticality, and Mandelbrot fractals

M

^{8}-H duality (see this) represents an intriguing connection between number theory and TGD but the mathematics involved is extremely abstract and difficult so that I can only represent conjectures. In the following the basic duality is used to formulate a general conjecture for the construction of preferred extremals by iterative procedure. What is remarkable and extremely surprising is that the iteration gives rise to the analogs of Mandelbrot fractals and space-time surfaces can be seen as fractals defined as fixed sets of iteration. The analogy with Mandelbrot set can be also seen as a geometric correlate for quantum criticality.

** M ^{8}-H duality**

M^{8}-H duality states the following. Consider a distribution of two planes M^{2}(x) integrating to a 2-surface N^{2} with the property that a fixed 1-plane M^{1} defining time axis globally is contained in each M^{2}(x) and therefore in N^{2}. M^{1} defines real axis of octonionic plane M^{8} and M^{2}(x) a local hyper-complex plane. Quaternionic subspaces with this property can be parameterized by points of CP_{2}. Define quaternionic surfaces in M^{8} as 4-surfaces, whose tangent plane is quaternionic at each point x and contains the local hyper-complex plane M^{2}(x) and is therefore labelled by a point s(x)∈ CP_{2}. One can write these surfaces as union over 2-D surfaces associated with points of N^{2}:

X^{4}= ∪_{x∈ N2} X^{2}(x)⊂ E^{6} .

These surfaces can be mapped to surfaces of M^{4}× CP_{2} via the correspondence (m(x),e(x))→ (m,s(T(X^{4}(x)). Also the image surface contains at given point x the preferred plane M^{2}(x) ⊃ M^{1}. One can also write these surfaces as union over 2-D surfaces associated with points of N^{2}:

X^{4}= ∪_{x∈ N2} X^{2}(x)⊂ E^{2}× CP_{2} .

One can also ask what are the conditions under which one can map surfaces X^{4}= ∪_{x∈ N2} X^{2}⊂ E^{2}× CP_{2} to 4-surfaces in M^{8}. The map would be given by (m,s)→ (m,T^{4}(s) and the surface would be of the form as already described. The surface X^{4} must be such that the distribution of 4-D tangent planes defined in M^{8} is integrable and this gives complicated integrability conditions. One might hope that the conditions might hold true for preferred extremals satisfying some additional conditions.

One must make clear that these conditions do not allow most general possible surface. The point is that for preferred extremals with Euclidian signature of metric the M^{4} projection is 3-dimensional and involves light like projection. Here the fact that light-like line L⊂ M^{2} spans M^{2} in the sense that the complement of its orthogonal complement in M^{8} is M^{2}. Therefore one could consider also more general solution ansatz for which one has

X^{4}= ∪_{x∈ Lx⊂ N2} X^{3}(x)⊂ E^{2}× CP_{2} .

One can also consider co-quaternionic surfaces as surfaces for which tangent space is in the dual of a quaternionic subspace containing the preferred M^{2}(x).

**The integrability conditions**

The integrability conditions are associated with the expression of tangent vectors of T(X^{4}) as a linear combination of coordinate gradients ∇ m^{k}, where m^{k} denote the coordinates of M^{8}. Consider the 4 tangent vectors e_{i)} for the quaternionic tangent plane (containing M^{2}(x)) regarded as vectors of M^{8}. e_{i)} have components e_{i)}^{k}, i=1,..,4, k=1,...,8. One must be able to express e_{i)} as linear combinations of coordinate gradients ∇ m^{k}:

e_{i)}^{k}= e_{i)}^{α}∂_{α}m^{k} .

Here x^{α} and e^{k} denote coordinates for X^{4} and M^{8}. By forming inner products of of e_{i)} one finds that matrix e_{i)}^{α} represents the components of vierbein at X^{4}. One can invert this matrix to get e^{i)}_{α} satisfying

e^{i)}_{α}e_{i)}^{β}=δ_{α}^{β}

and

e^{i)}_{α}e_{j)}^{α}=δ^{i}_{j}.

One can solve the coordinate gradients ∇ m^{k} from above equation to get

∂_{α}m^{k} = e^{i)}_{α}e_{i)}^{k}== E_{α}^{k} .

The integrability conditions follow from the gradient property and state

D_{α}E^{k}_{β}= D_{β}E^{k}_{α} .

One obtains 8× 6=48 conditions in the general case. The slicing to a union of two-surfaces labeled by M^{2}(x) reduces the number of conditions since the number of coordinates m^{k} reduces from 8 to 6 and one has 36 integrability conditions but still them is much larger than the number of free variables- essentially the six transversal coordinates m^{k}.

For co-quaternionic surfaces one can formulate integrability conditions now as conditions for the existence of integrable distribution of orthogonal complements for tangent planes and it seems that the conditions are formally similar.

**How to solve the integrability conditions and field equations for preferred extremals?**

The basic idea has been that the integrability condition characterize preferred extremals so that they can be said to be quaternionic in a well-defined sense. Could one imagine solving the integrability conditions by some simple ansatz utilizing the core idea of M^{8}-H duality? What comes in mind is that M^{8} represents tangent space of M^{4}× CP_{2} so that one can assign to any point (m,s) of 4-surface X^{4}⊂ M^{4}× CP_{2} a tangent plane T^{4}(x) in its tangent space M^{8} identifiable as subspace of complexified octonions in the proposed manner. Assume that s∈ CP_{2} corresponds to a fixed tangent plane containing M^{2}_{x}, and that all planes M^{2}_{x} are mapped to the same standard fixed hyper-octonionic plane M^{2}⊂ M^{8}, which does not depend on x. This guarantees that s corresponds to a unique quaternionic tangent plane for given M^{2}(x).

Consider the map Tοs. The map takes the tangent plane T^{4} at point (m,e)∈ M^{4}× E^{4} and maps it to (m,s_{1}=s(T^{4}))∈ M^{4}× CP_{2}. The obvious identification of quaternionic tangent plane at (m,s_{1}) would be as T^{4}. One would have Tοs=Id. One could do this for all points of the quaternion surface X^{4}⊂ E^{4} and hope of getting smooth 4-surface X^{4}⊂ H as a result. This is the case if the integrability conditions at various points (m,s(T^{4})(x))∈ H are satisfied. One could equally well start from a quaternionic surface of H and end up with integrability conditions in M^{8} discussed above. The geometric meaning would be that the quaternionic surface in H is image of quaternionic surface in M^{8} under this map.

Could one somehow generalize this construction so that one could iterate the map Tοs to get Tοs=Id at the limit? If so, quaternionic space-time surfaces would be obtained as limits of iteration for rather arbitrary space-time surface in either M^{8} or H. One can also consider limit cycles, even limiting manifolds with finite-dimension which would give quaternionic surfaces. This would give a connection with chaos theory.

- One could try to proceed by discretizing the situation in M
^{8}and H. One does not fix quaternionic surface at either side but just considers for a fixed m_{2}∈ M^{2}(x) a discrete collection X {T^{4}_{i}⊃ M^{2}(x) of quaternionic planes in M^{8}. The points e_{2,i}⊂ E^{2}⊂ M2× E^{2}=M^{4}are not fixed. One can also assume that the points s_{i}=s(T^{4}_{i}) of CP_{2}defined by the collection of planes form in a good approximation a cubic lattice in CP_{2}but this is not absolutely essential. Complex Eguchi-Hanson coordinates ξ^{i}are natural choice for the coordinates of CP_{2}. Assume also that the distances between the nearest CP_{2}points are below some upper limit.

- Consider now the iteration. One can map the collection X to H by mapping it to the set s(X) of pairs ((m
_{2},s_{i}). Next one must select some candidates for the points e_{2,i}∈ E^{2}⊂ M^{4}somehow. One can define a piece-wise linear surface in M^{4}× CP_{2}consisting of 4-planes defined by the nearest neighbors of given point (m_{2},e_{2,i},s_{i}). The coordinates e_{2,i}for E^{2}⊂ M^{4}can be chosen rather freely. The collection (e_{2,i},_{i}) defines a piece-wise linear surface in H consisting of four-cubes in the simplest case. One can hope that for certain choices of e_{2,i}the four-cubes are quaternionic and that there is some further criterion allowing to choose the points e_{2,i}uniquely. The tangent planes contain by construction M^{2}(x) so that the product of remaining two spanning tangent space vectors (e_{3},e_{4}) must give an element of M^{2}in order to achieve quaternionicity. Another natural condition would be that the resulting tangent planes are not only quaternionic but also as near as possible to the planes T^{4}_{i}. These conditions allow to find e_{2,i}giving rise to geometrically determined quaternionic tangent planes as near as possible to those determined by s^{i}.

- What to do next? Should one replace the quaternionic planes T
^{4}_{i}with geometrically determined quaternionic planes as near as possible to them and map them to points s_{i}slightly different from the original one and repeat the procedure? This would not add new points to the approximation, and this is an unsatisfactory feature.

- Second possibility is based on the addition of the quaternionic tangent planes obtained in this manner to the original collection of quaternionic planes. Therefore the number of points in discretization increases and the added points of CP
_{2}are as near as possible to existing ones. One can again determine the points e_{2,i}in such a manner that the resulting geometrically determined quaternionic tangent planes are as near as possible to the original ones. This guarantees that the algorithm converges.

- The iteration can be stopped when desired accuracy is achieved: in other words the geometrically determined quaternionic tangent planes are near enough to those determined by the points s
_{i}. Also limit cycles are possible and would be assignable to the transversal coordinates e_{2i}varying periodically during iteration. One can quite well allow this kind of cycles, and they would mean that e_{2}coordinate as a function of CP_{2}coordinates characterizing the tangent plane is many-valued. This is certainly very probable for solutions representable locally as graphs M^{4}→ CP_{2}. In this case the tangent planes associated with distant points in E^{2}would be strongly correlated which must have non-trivial physical implications. The iteration makes sense also p-adically and it might be that in some cases only p-adic iteration converges for some value of p.

^{4}× CP

_{2}a surface in M

^{8}. Therefore M

^{8}-H duality could express the integrability conditions and preferred extremals would be 4-surfaces having counterparts also in the tangent space M

^{8}of H.

One might hope that the self-referentiality condition sοT=Id for the CP_{2} projection of (m,s) or its fractal generalization could solve the complicated integrability conditions for the map T. The image of the space-time surface in tangent space M^{8} in turn could be interpreted as a description of space-time surface using coordinates defined by the local tangent space M^{8}. Also the analogy for the duality between position and momentum suggests itself.

Is there any hope that this kind of construction could make sense? Or could one demonstrate that it fails? If s would fix completely the tangent plane it would be probably easy to kill the conjecture but this is not the case. Same s corresponds for different planes M^{2}_{x} to different point tangent plane. Presumably they are related by a local G_{2} or SO(7) rotation. Note that the construction can be formulated without any reference to the representation of the imbedding space gamma matrices in terms of octonions. Complexified octonions are enough in the tangent space of M^{8}.

**Connection with Mandelbrot fractal and fractals as fixed sets for iteration**

The occurrence of iteration in the construction of preferred extremals suggests a deep connection with the standard construction of 2-D fractals by iteration - about which Mandelbrot fractal is the canonical example. X^{2}(x) (or X^{3}(x)) could be identified as a union of orbits for the iteration of sοT. The appearance of the iteration map in the construction of solutions of field equation would answer positively to a long standing question whether the extremely beautiful mathematics of 2-D fractals could have some application at the level of fundamental physics according to TGD.

X^{2} (or X^{3}) would be completely analogous to Mandelbrot set in the sense that it would be boundary separating points in two different basis of attraction. In the case of Mandelbrot set iteration would take points at the other side of boundary to origin on the other side and to infinity. The points of Mandelbrot set are permuted by the iteration. In the recent case sοT maps X^{2} (or X^{3}) to itself. This map need not be diffeomorphism or even continuous map. The criticality of X^{2} (or X^{3}) could be seen as a geometric correlate for quantum criticality.

In fact, iteration plays a very general role in the construction of fractals. Very general fractals can be defined as fixed sets of iteration and simple rules for iteration produce impressive representations for fractals appearing in Nature. Therefore it would be highly satisfactory if space-time surfaces would be in well-defined sense fixed sets of iteration. This would be also numerically beautiful aspect since fixed sets of iteration can be obtained as infinite limit of iteration for almost arbitrary initial set.

What is intriguing and challenging is that there are several very attractive approaches to the construction of preferred extremals and the challenge of unifying them still remains.

For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-Dimensional Geometry", or the article with the same title.

## 1 Comments:

http://arxiv.org/abs/hep-th/0603022

http://loops11.iem.csic.es/loops11/Archives/Parallel-Sessions/Sundance-Bilson-Thompson_Braided-topology-and-the-emergence-of-matter.pdf

http://arxiv.org/pdf/1109.0080.pdf

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