In TGD, space-time surface can be regarded as a 4-D root for a pair P

_{1},P

_{2} of polynomials of generalized complex coordinates of H=M

^{4}× CP

_{2} (of of the coordinates is generalized complex coordinates varying along light-like curves). Each pair gives rise to a 6-D surface proposed to be identifiable as analog of twistor space and their intersection defines space-time surface as a common base of these twistor spaces as S

^{2}.

One can also think of the space-time surface X^{4} as a base space of a twistor surface X^{6} in the product T(M^{4})× T(CP_{2}) of the twistor spaces of M^{4} and H. One can represent X^{4} as a section of this twistor space as a root of a single polynomial P. The number roots of a polynomial does not depend on the point chosen. One considers polynomials with rational coefficients but also analytic functions can be considered and general coordinate invariance would suggest that they should be allowed.

Could one generalize the notion of the Galois group so that one could speak of a Galois group acting on 4-surface X^{4} permuting its sheets as roots of the polynomial? Could one speak of a local Galois group with local groups Gal(x) assigned with each point x of the space-time surface. Could one have a general coordinate invariant definition for the generalized Galois group, perhaps working even when one considers analytic functions f_{1},f_{2} instead of polynomials. Also a general coordinate invariant definition of ramified primes identifiable as p-adic primes defining the p-adic length scales would be desirable.

The required view of the Galois group would be nearer to the original view of Galois group as permutations of the roots of a polynomial whereas the standard definition identifies it as a group acting as an automorphism in the extension of the base number field induced by the roots of the polynomial and leaving the base number field. The local variant of the ordinary Galois group would be defined for the points of X^{4} algebraic values of X^{4} coordinates and would be trivial for most points. Something different is needed.

In the TGD framework, a geometric realization for the action of the Galois group permutings space-time regions as roots of a polynomial equation is natural and the localization of the Galois group is natural. I have earlier considered a realization as a discrete subgroup of a braid group which is a covering group of the permutation group. The twistor approach leads to an elegant realization as discrete permutations of the roots of the polynomial as values of the S^{2} complex coordinate of the analog of twistor bundle realized as a 6-surface in the product of twistor spaces of M^{4} and CP_{2}. This realization makes sense also for the P_{1},P_{2} option.

The natural idea is that the Galois group acts as conformal transformations or even isometries of the twistor sphere S^{2}. The isometry option leads to a connection with the McKay correspondence. Only the Galois groups appearing in the hierarchy finite subgroups of rotation groups appearing in the hierarchy of Jones inclusions of hyper-finite factors of type II_{1} are realized as isometries and only the isometry group of the cube is a full permutation group. For the conformal transformations the situation is different. In any case, Galois groups representable as isometries of S^{2} are expected to be physically very special so that the earlier intuitions seems to be correct.

General coordinate invariance allows any coordinates for the space-time surface X^{4} as the base space of X^{6} as the analog of twistor bundle and the complex coordinate z of S^{2} is determined apart from linear holomorphies z → az+b, which do not affect the ramimifed primes as factors of the discriminant defined by the product of the root differences.

See the article TGD as it is towards end of 2024: part I
or a 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.