A fresh look at M8-H duality and Poincare invariance

M⁸-H duality is a proposal to integrate geometric and number theoretic visions of TGD. M⁸-H duality has several questionable features. For various reasons it seems that M⁸ must be replaced with its complexification M⁸_c interpreted as complexified octonions O_c. This however leads to several problems. The modified variant of M⁸-H duality identifying M⁸ as a quaternionic sub-space of octonions O with a number theoretic norm defined by Re(o²), rather than oo*, solves these problems.

The proposal has been that octonionic polynomials P(o) define the number theoretic holography. Their roots would define 3-D mass shells for which mass squared values are in general complex and the initial data for the holography would correspond to 3-surfaces at these mass shells. Also this assumption has problems. There is however no need for this assumption: the holography on the H side is induced by the M⁸-H duality!

The hierarchy of polynomials defines a hierarchy of algebraic extensions defining an evolutionary hierarchy central for all applications of TGD and one must have it. Luckily, the recent realization that a generalized holomorphy realizes the holography at the H side as roots for pairs of holomorphic functions of complex (in generalized sense) coordinates of H comes to rescue. It can be strengthened by assuming that the functions form a hierarchy of pairs of polynomials.

Twistor lift strongly suggests that M⁴ and space-time surfaces allow a Kähler structure and what I call Hamilton-Jacobi structure. These structures force a breaking of Poincare and even Lorentz invariance unless they are dynamically generated. It indeed turns out that M⁸-H duality generates them dynamically.

See the article A fresh look at M⁸-H duality and Poincare invariance or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.