About Langlands program
The following represents an off-topic comment to Not-Even-Wrong which I never even considered to post. Peter Woit has a nice posting about Langlands program about which I wrote a chapter for year or so ago with emphasis on what I call number theoretic braids, on the identification of Galois group Gal of algebraic numbers as a permutation group S∞ for infinite number of objects, and on the connection with hyper-finite factors of type II1 emerging via the fact that the group algebra of S∞ is HFF of type II1.
This background perhaps explains why I had several fleeting impressions of "At some primitive level I might understand this!" while reading Peter's summary. The question about the physical interpretation of the constructs of Geometric Langlands was raised but Peter made with an admirable clarity clear that any comments relating to fundamental physics by non-names would be deleted.
1. Some ideas related to number theoretic Langlands program and their counterparts in TGD framework
The first nice idea of the number theoretic Langlands program is that rationals can be formally regarded as "rational functions" in a discrete space of primes and that the extensions of rationals can be regarded as covering spaces of this "function space" characterized by Galois groups.
In TGD framework this analogy appears in the reverse direction. Infinite rationals and their algebraic extensions (in particular infinite primes) forming an infinite hierarchy can be constructed by an iterated second quantization of an arithmetic quantum field theory as quantum states of a generalized arithmetic QFT. The reverse of analogy means that infinite rationals can be mapped to rational functions.
Langlands program has local and global aspects which in TGD framework would physically correspond to various p-adic physics and real physics (p-adic physics would relate to cognition and intentionality). Note however that p-adic space-time sheets have literally infinite size in the real sense (rational points are common to real and p-adic space-time sheets). Local aspect means that to each prime one can assign a local function field and it corresponds to a p-adic number field Qp. The elements of Qp are analogs of formal Laurent series which in general do not converge in real sense. The local Langlands conjecture gives a correspondence between the representations of Gal(Qp) into a complex Lie group G and complex representations of the corresponding algebraic group LG(Qp) with Qp coefficients. The global aspect is about Gal(Q) and Langlands relates the representations of Gal(Q) in group G and the dual group LG (AQ), where AQ denotes adeles.
2. Geometric Langlands program
Witten and others are developing geometric Langlands program which is a generalization of Langlands program from number fields to holomorphic function fields defined at 2-dimensional Riemann surfaces. Now duality corresponds to electric-magnetic duality originally conjectured by Olive and Montonen that one can assign to a gauge theory dual gauge theory formulated in terms of magnetic charges regarded as gauge charges of the dual group LG instead of electric charges in G. It would be interesting to relate G and its dual to the groups related to the inclusions of HFFs.
Conformal field theory is a central element of the approach. In particular, disks with punctures appear as a basic notion. Some kind of 4-D theory (twisted 4-D SYM) giving rise to Chern-Simons theory giving rise to 2-D conformal field theory is conjectured to provide representations for groups and their duals.
3. Quantum theory according to TGD
Before comparison with TGD I want to emphasize that quantum theory according to TGD differs from standard quantum theory in many respects.
- TGD can be formulated using several philosophies about what quantum physics is: quantum physics as Kähler geometry and spinor structure for the world of classical worlds; quantum physics as almost topological QFT; quantum physics as generalized number theory with associativity defining the fundamental dynamical principle; quantum physics reduced to the notion of measurement resolution formulated in terms of inclusions for HFFs of type II1 (or possibly more general algebras).
- Kähler geometry of the world of classical worlds provide a geometrization of quantum theory. Kähler function as a functional of light-like 3-surface is defined as Kähler action for the preferred extremal. Fermionic oscillator operators are building blocks of gamma matrices of this space and define super-generators of super-canonical algebra.
- Zero energy ontology implies radical deviation from the standard QFT framework. S-matrix is generalized to M-matrix defined as the complex square root of density matrix with unitary S-matrix appearing as analog of phase factor. The idea is that quantum theory is square root of statistical physics. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state having interpretation as initial and final states of particle reaction.
- Measurement resolution is realized in terms of inclusions of HFFs of type II1 leading to a highly unique identification of M-matrix in terms of Connes tensor product. The non-uniqueness corresponds to statistical physics degrees of freedom. M-matrix has Hermitian operators of the included algebra defining measurement resolution as symmetries. These symmetries should include super-conformal symmetries.
- Hierarchy of Planck constants realized in terms of book like structure of the imbedding space and introduction of p-adic copies of imbedding space glued to real imbedding space along common rational and algebraic points mean radical generalization of the standard quantum theory framework. Hierarchy of Planck constants brings in dark matter as a hierarchy of macroscopically quantum coherent phases and p-adic physics gives rise to the correlates of intentionality and cognition.
Geometric Langlands program brings strongly to my uneducated mind TGD as an almost topological QFT with generalized conformal symmetries (light-like 3-surfaces are metrically 2-dimensional). 4-D twisted SYM would be replaced with Kähler action plus holography in the restricted sense that 4-D physics provides a (non-faithful) classical representation of the fundamental 3-D lightlike partonic quantum physics in terms of the preferred extrema of Kähler action having identification as generalized Bohr orbits. This is necessary for quantum measurement theory to make sense at the fundamental level.
Punctures, whose interpretation remains unclear in the Langlands program, associate themselves naturally with the intersections of the initial and final partonic 2-surface of particle reaction with the generalized number theoretic braids (strands can now fuse) defined by the orbits of the minima of Higgs expectation identified as a generalized eigenvalue of the modified Dirac operator (certain holomorphic function on partonic 2-surface by the properties of modified Dirac). Partons correspond to quantum states created by applying fermionic fields at the points of the number theoretic braid so that one has a concrete physical interpretation. TGD as a generalized number theory vision of course relies on the idea that fundamental physics provides a representation for number theory understood in some very general sense.
Physically punctures correspond to the lowest step in a dimensional hierarchy. Light-like 3-surface decomposes to cells bounded by 2-D surfaces such that each 3- region is independent dynamical unit. One has effectively discretized 3-D physics. The resulting 2-D surfaces (partons) obey conformal field theory separately and a collection of 1-D curves serves as causal determinants for them. Number theoretic universality in turn forces to select Higgs minima as a subset of points common to real and p-adic partonic 2-surfaces.
What is interesting that the number theoretical braids emerge in the TGD based proposal for the formulation of the number theoretic Langlands program based on Gal(Q)= S∞ identification and relates also naturally to the conformal field approach appearing in the geometric Langlands. Could number theoretic braids allow to unify number theoretic and geometric Langlands as the unification of number fields and rational function fields provided by the notion of infinite prime suggests? This I cannot of course answer. These are just ideas inspired by the physics of TGD. It would be flattering if some real mathematician would consider these ideas seriously. I dare to believe that since TGD seems to be a working physical theory it could help also to discover "working" mathematics.