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New insights about Langlands duality

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Gary Ehlenberg sent an URL of a very interesting QuantaMagazine article, which discusses a work related to Langlands program.

Langlands duality relates number theory and geometry. At the number theory side one has representations of Galois groups. On the geometry side one has automorphic forms associated with the representations of Lie groups. For instance, in coset spaces of hyperbolic 3-space H3 in the case of the Lorentz group.

The work could be highly interesting from the TGD perspective. In TGD, the M8-H duality generalizes momentum-position duality so that it applies to particles represented as 3-surfaces instead of points. M8-H duality also relates physics as number theory and physics as geometry. Much like Langlands duality. The problem is to understand M8-H duality as an analog of Langlands duality.

  1. H=M4×CP2 is the counterpart of position space and particle corresponds to 3-surface in H. Physics as (differential) geometry applies at this side.

The orbit of 3-surface is a 4-D space-time surface in H and holography, forced by 4-D general coordinate invariance, implies that space-time surfaces are minimal surfaces irrespective of the action (general coordinate invariant and determined by induced geometry) . They would obey 4-D generalization of holomorphy and this would imply universality.

These minimal surfaces are also solutions of a nonlinear geometrized version of massless field equations. Field-particle duality has a geometrized variant: minimal surface represents in its interior massless field propagation and as an orbit of 3-D particles the generalization of a light-like geodesic. Hence a connection with electromagnetism mentioned in the popular article, actually metric and all gauge fields of the standard model are geometrized by induction procedure for geometry.

  • M8, or rather its complexification M8c (complexification is only with respect to mass squared as coordinate,not hyperbolic and other angles) corresponds to momentum space and here the orbit of point-like particle in momentum space is replaced with a 4-surface in M8, or actually its complexification M8c.
  • The 3-D initial data for a given extension of rationals could correspond to a union of hyperbolic 3-manifolds as a union of fundamental regions for a tessellation of H3 consistent with the extensions, a kind of hyperbolic crystal. These spaces relate closely to automorphic functions and L-functions.

    At the M8 side polynomials with rational coefficients determine partially the 3-D data associated with number theoretical holography at M8-side. The number theoretical dynamical principle states that the normal space of the space-time surface in the octonionic M8c is associative and initial data correspond to 3-surfaces at mass shells H3c ⊂ M4c ⊂ M8c determined by the roots of the polynomial.

  • M8-H duality maps the 4-surfaces in M8c to space-time surfaces in H. At the M8 side one has polynomials. At the geometric H-side one has naturally the generalizations of periodic functions since Fourier analysis or its generalization is natural for massless fields which space-time surfaces geometrize. L-functions represent a typical example of generalized periodic functions.
  • M8-H duality is a precisely defined notion but the problem is to deduce explicit formulas between the representations of 4-surfaces at M8 and H-side. At the M8 side one has polynomials and roots and at the H-side one has automorphic functions and periods. How do they relate? The popular article suggests that Ben-Zvi, Sakellaridis and Venkatesh are basically working with the same problem which troubles me and for which I cannot of course do anything! See for instance the article Some New Ideas Related to Langlands Program viz TGD.

    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.


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