Wednesday, July 27, 2016

Langlands program and TGD

Langlands correspondence is for mathematics what unified theories are for physics. The number theoretic vision about TGD has intriguing resemblances with number theoretic Langlands program. There is also geometric variant of Langlands program. I am of course amateur and do not have grasp about the mathematical technicalities and can only try to understand the general ideas and related them to those behind TGD. Physics as geometry of WCW ("world of classical worlds") and physics as generalized number theory are the two visions about quantum TGD: this division brings in mind geometric and number theoretic Langlands programs. This motivates re-consideration of Langlands program from TGD point of view. I have written years ago a chapter about this earlier but TGD has evolved considerably since then so that it is time for a second attempt to understand what Langlands is about.

By Langlands correspondence the representations of semi-direct product of G and Galois group Gal and G should correspond to each other. This suggests that he representations of G should have G-spin such that the dimension of this representation is same as the representation of non-commutative Galois group. This would conform with the vision about physics as generalized number theory. Could this be the really deep physical content of Langlands correspondence?

See the chapter Langlands correspondence and TGD: years later of "Physics as generalized number theory" or the article with the same title.

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

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