TGD leads to several proposals for the exact solution of field equations defining space-time surfaces as preferred extremals of twistor lift of Kähler action. So called M8-H duality is one of these approaches. The beauty of M8-H duality is that it could reduce classical TGD to algebraic geometry and would immediately provide deep insights to cognitive representation identified as sets of rational points of these surfaces.
In the sequel I shall consider the following topics.
- I will discuss basic notions of algebraic geometry such as algebraic variety, surface, and curve, rational point of variety central for TGD view about cognitive representation, elliptic curves and surfaces, and rational and potentially rational varieties. Also the notion of Zariski topology and Kodaira dimension are discussed briefly. I am not a mathematician and what hopefully saves me from horrible blunders is physical intuition developed during 4 decades of TGD.
- It will be shown how M8-H duality could reduce TGD at fundamental level to algebraic geometry. Space-time surfaces in M8 would be algebraic surfaces identified as zero loci for imaginary part IM(P) or real part RE(P) of octonionic polynomial of complexified octonionic variable oc decomposing as oc= q1c+q2c I4 and projected to a Minkowskian sub-space M8 of complexified O. Single real valued polynomial of real variable with algebraic coefficients would determine space-time surface! As proposed already earlier, spacetime surfaces would form commutative and associative algebra with addition, product and functional composition.
One can interpret the products of polynomials as correlates for free many-particle states with interactions described by added interaction polynomial, which can vanish at boundaries of CDs thanks to the vanishing in Minkowski signature of the complexified norm qcqc* appearing in RE(P) or IM(P) caused by the quaternionic non-commutativity. This leads to the same picture as the view about preferred extremals reducing to minimal surfaces near boundaries of CD. Also zero zero energy ontology (ZEO) could emerge naturally from the failure of number field property for for quaternions at light-cone boundaries.
- The fundamental challenge is to prove that the octonionic polynomials with real coefficients determine associative/quaternionic surfaces as the zero loci of their imaginary/real parts in quaternionic sense. Here the intuition comes from the idea that the octonionic polynomials map from octonionic space O to second octonionic space W. Real and imaginary parts in W are quaternionic/co-quaternionic. These planes correspond to surfaces in O defined by the vanishing of real/imaginary parts, and the natural guess is that they are quaternionic/co-quaternionic, that is associative/co-associative.
The hierarchy of notions involved is well-ordering for 1-D structures, commutativity for complex numbers, and associativity for quaternions. This suggests a generalization of Cauchy-Riemann conditions for complex analytic functions to quaternions and octonions. Cauchy Riemann conditions are linear and constant value manifolds are 1-D and thus well-ordered. Quaternionic polynomials with real coefficients define maps for which the 2-D spaces corresponding to vanishing of real/imaginary parts of the polynomial are complex/co-complex or equivalently commutative/co-commutative. Commutativity is expressed by conditions bilinear in partial derivatives. Octonionic polynomials with real coefficients define maps for which 4-D surfaces for which real/imaginary part are quaternionic/co-quaternionic, or equivalently associative/co-associative. The conditions are now 3-linear.
In fact, all algebras obtained by Cayley-Dickson construction adding imaginary units to octonionic algebra are power associative so that polynomials with real coefficients define an associative and commutative algebra. Hence octonion analyticity and M8-H correspondence could generalize.
- It turns out that in the generic case associative surfaces are 3-D and are obtained by requiring that one of the coordinates RE(Y)i or IM(Y)i in the decomposition Yi=RE(Y)i +IM(Y)iI4 of the gradient of RE(P)= Y=0 with respect to the complex coordinates zik, k=1,2, of O vanishes that is critical as function of quaternionic components z1k or z2k associated with q1 and q2 in the decomposition o= q1+q2I4, call this component Xi. In the generic case this gives 3-D surface.
In this generic case M8-H duality can map only the 3-surfaces at the boundaries of CD and light-like partonic orbits to H, and only determines the boundary conditions of the dynamics in H determined by the twistor lift of Kähler action. M8-H duality would allow to solve the gauge conditions for SSA (vanishing of infinite number of Noether charges) explicitly.
One can also have criticality. 4-dimensionality can be achieved by posing conditions on the coefficients of the octonionic polynomial P so that the criticality conditions do not reduce the dimension: Xi would have possibly degenerate zero at space-time variety. This can allow 4-D associativity with at most 3 critical components Xi. Space-time surface would be analogous to a polynomial with a multiple root. The criticality of Xi conforms with the general vision about quantum criticality of TGD Universe and provides polynomials with universal dynamics of criticality. A generalization of Thom's catastrophe theory emerges. Criticality should be equivalent to the universal dynamics determined by the twistor lift of Kähler action in H in regions, where Kähler action and volume term decouple and dynamics does not depend on coupling constants.
One obtains two types of space-time surfaces. Critical and associative (co-associative) surfaces can be mapped by M8-H duality to preferred critical extremals for the twistor lift of Kähler action obeying universal dynamics with no dependence on coupling constants and due to the decoupling of Kähler action and volume term: these represent external particles. M8-H duality does not apply to non-associative (non-co-associative) space-time surfaces except at 3-D boundary surfaces. These regions correspond to interaction regions in which Kähler action and volume term couple and coupling constants make themselves visible in the dynamics. M8-H duality determines boundary conditions.
- Cognitive representations are identified as sets of rational points for algebraic surfaces with "active" points containing fermion. The representations are discussed at both M8- and H level. Rational points would be now associated with 4-D algebraic varieties in 8-D space. General conjectures from algebraic geometry support the vision that these sets are concentrated at lower-dimensional algebraic varieties such as string world sheets and partonic 2-surfaces and their 3-D orbits, which can be also identified as singularities of these surfaces.
- Some aspects related to homology charge (Kähler magnetic charge) and genus-generation correspondence are discussed. Both are central in the proposed model of elementary particles and it is interesting to see whether the picture is internally consistent and how algebraic surface property affects the situation. Also possible problems related to heff/h=n hierarchy realized in terms of n-fold coverings of space-time surfaces are discussed from this perspective.
Octonions and quaternions are 20 year old part of TGD: one of the three threads in physics as generalized number theory vision. Second vision is quantum physics as geometry of WCW. The question has been how to fuse geometric and number theory visions. Algebraic geometry woul do it since it is both geometry and algebra and it has been also part of TGD but only now I realized how to get acceess to its enormous power.
Even the proposal discussed now about the algebra of octonionic polynomials with real coefficients was made about two decades ago but only now I managed to formulate it in detail. Here the general wisdom gained from adelic physics helped enormously. I dare say that classical TGD at the most fundamental level is solved exactly.
From the point of pure mathematics the generalization of complex analyticity and linear Cauchy Riemann conditions to multilinear variants for quaternions, octonions and even for the entire hierarchy of algebras obtained by Cayley-Dickson construction is a real breakthrough. Consider only the enormous importance of complex analyticity in mathematics and physics, including string models. I do not believe that this generalization has been discovered: otherwise it would be key part of mathematical physics. Quaternionic and octonionic analyticities will certainly mean huge evolution in mathematics. I had never ended to these discoveries without TGD: TGD forced them.
At these moments I feel deep sadness when knowing that the communication of these results to collegues is impossible in practice. This stupid professional arrogance is something which I find very difficult to accept even after 4 decades. I feel that when society pays a monthly salary for a person for being a scientists, he should feel that his duty is to be keenly aware what is happening in his field. When some idiot proudly tells that he reads only prestigious journals, I get really angry.
For details see the article Does M8-H duality reduce classical TGD to octonionic algebraic geometry?or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.
For a summary of earlier postings see Latest progress in TGD.