Sunday, July 07, 2024

Do   local Galois group and ramified primes make sense as general coordinate invariant notions?

In TGD, space-time surface can be regarded as a 4-D root for a pair P1,P2 of polynomials of generalized complex coordinates of H=M4× CP2 (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 S2.

One can also think of the space-time surface X4 as a base space of a twistor surface X6 in the product T(M4)× T(CP2) of the twistor spaces of M4 and H. One can represent X4 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 X4 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 f1,f2 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 X4 algebraic values of X4 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 S2 complex coordinate of the analog of twistor bundle realized as a 6-surface in the product of twistor spaces of M4 and CP2. This realization makes sense also for the P1,P2 option.

The natural idea is that the Galois group acts as conformal transformations or even isometries of the twistor sphere S2. 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 II1 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 S2 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 X4 as the base space of X6 as the analog of twistor bundle and the complex coordinate z of S2 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.

No comments: