What makes me happy is that TGD is not only receiving experimental support from LHC and other particle accelerators but also providing profound insights inspiring mathematical conjectures. What is also highly satisfying that the physically motivated visions such as the need for number theoretic universality are guiding the development of modern mathematics. The notion of motives introduced by Grothendieck is a good example about number theoretical universality and relates to the need to define integral- at least in cohomological sense- for all number fields: very concrete challenge also in TGD framework.
Langlands program is one of the hot areas of what might be called physical mathematics. The above mentioned number theoretical universality is one of the guiding lines in this approach. The program relies on very general conjectures about a connection between number theory and harmonic analysis relating the representations of Galois groups with the representations of certain kinds of Lie groups to each other. Langlands conjecture has many forms and it is indeed a conjecture and many of them are inprecise since the notions involved are not sharply defined.
Peter Woit noticed that Ed Frenkel had given a talk with rather interesting title "What do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?"? I listened the talk and found it very inspiring. The talk provides bird's eye of view about some basic aspects of Langlands program using the language understood by physicist. Also the ideas concerting the connection between Langlands duality and electric-magnetic duality generalized to S-duality in the context of non-Abelian gauge theories and string theory context and developed by Witten and Kapustin and followers are summarized. In this context D=4 and twisted version of N=4 SYM familiar from twistor program and defining a topological QFT appears.
For some years ago I made my first attempt to understand what Langlands program is about and tried to relate it to TGD framework (see this). At that time I did not really understand the motivations for many of the mathematical structures introduced. In particular, I did not really understand the motivations for introducing the gigantic Galois group of algebraic numbers regarded as algebraic extension of rationals.
- Why not restrict the consideration to finite Galois groups or their braided counterparts (as I indeed effectively did in my first approach)? At that time I concentrated on the question what enormous Galois group of algebraic numbers regarded as algebraic extension of rationals could mean, and proposed that it could be identified as a symmetric group consisting of permutations of infinitely many objects. The definition of this group is however far from trivial. Should one allow as generators of the group only the permutations affecting only finite number of objects or permutations of even infinite number of objects?
The analogous situation for the sequences of binary digits would lead to a countable set of sequence of binary digits forming a discrete set of finite integers in real sense or to 2-adic integers forming a 2-adic continuum. Something similar could be expected now. The physical constraints coming the condition that the elements of symmetric group allow lifting to braidings suggested that the permutations permuting infinitely many objects should be periodic meaning that the infinite braid decomposes to an infinite number of identical N-braids and braiding is same for all of them. The p-adic analog would be p-adic integers, which correspond to rationals having periodical expansion in powers of p. Braids would be therefore like pinary digits. I regarded this choice as the most realistic one at that time. I failed to realize the possibility of having analogs of p-adic integers by general permutations. In any case, this observation makes clear that the unrestricted Galois group is analogous to a Lie group in topology analogous to p-adic topology rather than to discrete group. Neither did I realize that the Galois groups could be finite and be associated with some other field than rationals, say a Galois group associated with the field of polynomials of n-variable with rational coefficients and with its completion with coefficients replaced by algebraic numbers.
- The ring of adeles can be seen as a Cartesian product of non-vanishing real numbers R× with the infinite Cartesian product ∏ Zp having as factors p-adic integers Zp for all values of prime p. Rational adeles are obtained by replacing R with rationals Q and requiring that multiplication of rational by integers is equivalent with multiplication of any Zp with rational. Finite number of factors in Zp can correspond to Qp: this is required in to have finite adelic norm defined as the product of p-adic norms. This definition implicitly regards rationals as common to all number fields involved. At the first encounter with adeles I did not realize that this definition is in spirit with the basic vision of TGD.
The motivation for the introduction of adele is that one can elegantly combine the algebraic groups assignable to rationals (or their extensions) and all p-adic number fields or even more general function fields such as polynomials with some number of argument at the same time as a Cartesian product of these groups as well as to finite fields. This is indeed needed if one wants to realize number theoretic universality which is basic vision behind physics as generalized number theory vision. This approach obviously means enormous economy of thought irrespective of whether one takes adeles seriously as a physicist.
The talk of Frenkel inspired me to look again for Langlands program in TGD framework taking into the account of various developments that have occured in TGD during these years. I realized again that ideas develop unconsciously during the years and that many questions which remained unanswered for some years ago had found obvious answers. Instead of writing a 10 page posting I attach the abstract of pdf article "Langlands conjectures in TGD framework" at my homepage.
The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.
- Zero energy ontology and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M4 and of CP2). This notion leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet.
- Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.
- The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II1 at quantum level and represented in terms of confining effective gauge group (see this). This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations ("symmetry group" hitherto). At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.
There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable.
In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD.
- The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M4 coordinates for the braid strands.
- Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also the effective symmetry group has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.
- The symplectic group of the light-cone boundary acting as isometries of the WCW geometry allowing to represent projectively both Galois groups and effective symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symetry group.
- The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.