How to assign ordinary Galois groups and ramified primes to the space-time surfaces in holography=holomorphy vision?

Holography = holomorphy vision allows to reduce the construction of space-time surfaces as roots of a pair (f₁,f₂) of analytic functions of one hypercomplex coordinate and 3 complex coordinates of H=M⁴× CP₂. This allows iteration as the basic operation of chaos theory. One can consider general maps (f₁,f₂)→ G(f₁,f₂) =(g₁(f₁,f₂),g₂(f₁,f₂)) and iterate them. The special case (g₁(f₁,f₂),g₂(f₁,f₂))=(g₁(f₁),g₂(f₂)) gives iterations of functions g_i of a single complex variable appearing in the construction of Mandelbrot and Julia fractals.

Extensions of rational, Galois groups, and ramified primes assignable to polynomials of a single complex variable are central in the number theoretic vision. It is not however completely clear how they should emerge from the holography= holomorphy vision.

1. If the functions g_i== P_i are polynomials, which vanish at the origin (0,0) (this is not a necessary condition), the surfaces (f₁,f₂)=(0,0) are roots of (P₁(f₁,f₂),P₂(P₁,f₂))=(0,0). Besides these roots, there are roots for which (f₁,f₂) does not vanish. One can solve the roots f₂= h(f₁) from g₂(f₁,f₂) =0 and substitute to P₁(f₁,f₂)=0 to get P₁(f₁,h(f₁))== P_1° H(f₁))=0. The values of H(f₁) are roots of P₁ and are algebraic numbers if the coefficients of P₁ are in an extension of rationals. One can assign to the roots discriminant, ramified primes, and Galois group. This is just what the phenomenological number theoretical picture requires.
2. In the earliest approach to M⁸-H duality summarized in (see this, this, and this) polynomials P of a single complex coordinate played a key role. Although this approach was a failure, it added to the number theoretic vision Galois groups and ramified primes as prime factors of the discriminant P, identified as p-adic primes in p-adic mass calculations. Note that in the general case the ramified primes are primes of algebraic extensions of rationals: the simplest case corresponds to Gaussian primes and Gaussian Mersenne primes indeed appear in the applications of TGD (see this and this).

The problem was how to assign a Galois group and ramified primes to the space-time surfaces as 4-D roots of (f₁,f₂)=(0,0). One can indeed define the counterpart of the Galois group defined as analytic flows permuting various 4-D roots of (f₁,f₂)=(0,0) (see this). Since the roots are 4-D surfaces, it is far from clear whether there exists a definition of discriminant as an analog for the product of root differences. Also it is unclear what the notion of prime could mean.

However, the ordinary Galois group plays a key role in the number theoretic vision: can one identify it? The physics inspired proposal has been that the ordinary Galois group can be assigned to the partonic 2-surfaces so that points of the partonic 2-surface as roots of a polynomial give rise to the Galois group and ramified primes. An alternative identification of the ordinary Galois group and ramified primes would be in terms of (P₁(f₁,f₂),g₂(P₁,f₂))=(0,0). See the article Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness 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.