New insights about Langlands duality

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.

2. M⁸, or rather its complexification M⁸_c (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 M⁸_c.

   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 M⁸_c is associative and initial data correspond to 3-surfaces at mass shells H³_c ⊂ M⁴_c ⊂ M⁸_c determined by the roots of the polynomial.

3. M⁸-H duality maps the 4-surfaces in M⁸_c 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. Are the space-time surfaces at H-side expressible in terms of modular function in H³?

Here one must stop and take a breath. There are reasons to be very cautious! The proposed general exact solution of space-time surfaces as preferred extremals realizing almost exact holography as analogs of Bohr orbits of 3-D surfaces representing particles relies on a generalization of 2-D holomorphy to its 4-D analog. The 4-D generalization of holomorphic functions (see for instance this) assignable to 4-surfaces in H do not correspond to modular forms in 3-D hyperbolic manifolds assignable to the fundamental regions of tessellations of hyperbolic 3-space H³ (analogs of lattice cells in E³). Fermionic holography reduces the description of fermion states as wave functions at the mass shells of H³ and their images in H under M⁸-H duality, which are also hyperbolic 3-spaces.

1. This brings the modular forms of H³ naturally into the picture. Single fermion states correspond to wave functions in H³ instead of E³ as in the standard framework replacing infinite-D representations of the Poincare group with those of SL(2,C). The modular forms defining the wave functions inside the fundamental region of tessellation of H³ are analogs of wave functions of a particle in a box satisfying periodic boundary conditions making the box effectively a torus. Now it is replaced with a hyperbolic 3-manifold. The periodicity conditions code invariance under a discrete subgroup Γ of SL(2,C) and mean that H³ = SL(2,C)/U(2) is replaced with the double coset space Γ\SL(2,C)/U(2).

   Number theoretical vision makes this picture more precise and suggests ideas about the implications of the TGD counterpart of the Langlands duality.

2. Number theoretical approach restricts complex numbers to an extension of rationals. The complex numbers defining the elements SL(2,C) and U(2,C) matrices are replaced with matrices in discrete subgroups SL(2,F) and U(2,F), where F is the extension of rationals associated with the polynomial P defining the number theoretical holography in M⁸ inducing holography in H by M⁸-H duality. The group Γ defining the periodic boundary conditions must consist of matrices in SL(2,F).
3. The modular forms in H³ as wave functions are labelled by parameters analogous to momenta in the case of E³: in the case of E³ they characterize infinite-D irreducible representations of SL(2,C) as covering group of SO(1,3) with partial waves labelled by angular momentum quantum numbers and spin and by the analog of angular momentum associated with the hyperbolic angle (known as rapidity in particle physics): infinitesimal Lorentz boost in the direction of spin axis.

   The irreps are characterized by the values of a complex valued Casimir element of SL(2,C) quadratic in 3 generators of SL(2,C) or equivalently by two real Casimir elements of SO(1,3). Physical intuition encourages the shy question whether the second Casimir operator could correspond to the complex mass squared value defining the mass shell in M⁸. It belongs to the extension of rationals considered as a root of P.

   The construction of the unitary irreps of SL(2,C) is discussed in Wikipedia article. The representations are characterized by half integer j₀=n/2 and imaginary real number j₁= iν.

   The values of j₀ and j₁ must be restricted to the extension of rationals associated with the polynomial P defining the number theoretic holography.

4. The Galois group of the extension acts on these quantum numbers. Angular momentum quantum numbers are quantized already without number theory and are integers but the action on the hyperbolic momentum is of special interest. The spectrum of hyperbolic angular momentum must consist of a union of orbits of the Galois group and one obtains Galois multiplets. The Galois group generates from an irrep with a given value of j₁ a multiplet of irreps.

   A good guess is that the Galois action is central for M⁸-H duality as a TGD analog of Langlands correspondence. The Galois group would act on the parameter space of modular forms in Γ\SL(2,F)/U(2,F), F and extension of complex rationals and give rise to multiplets formed by the irreps of SL(2,F).

To sum up, M⁸-H duality is a rather precisely defined notion.

1. At the M⁸ side one has polynomials and roots and at the H-side one has automorphic functions in H³ and "periods" are interpreted as quantum numbers. What came first in my mind was that understanding of M⁸ duality boils down to the question about how the 4-surfaces given by number theoretical holography as associativity of normal space relate to those given by holography (that is generalized holomorphy) in H.
2. However, it seems that the problem should be posed in the fermionic sector. Indeed, above I have interpreted the problem as a challenge to understand what constraints the Galois symmetry on M⁸ side poses on the quantum numbers of fermionic wave functions in hyperbolic manifolds associated with H³. I do not know how closely this problem relates to the problem that Ben-Zvi, Sakellaridis and Venkatesh have been working with.

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.