## Sunday, June 25, 2017

### About McKay and Langlands correspondences in TGD framework

In adelic TGD Galois groups for extensions of rationals become discrete symmetry groups acting on dark matter, identified as heff/h=n phases of ordinary matter. n gives the number of sheet of covering assignable to space-time surface. Since Galois group acts on the cognitive representation defined by a discrete set of points of space-time surface with coordinates having values in extension of rationals, the action of Galois group defines n-sheeted covering, where n is the order of Galois group thus identifiable in terms of Planck constant.

Adelic TGD inspires the question whether the representations of Galois groups could correspond to representations of Lie groups defining the ground states of Kac-Moody representations emerging in TGD in two manners: as representations of Kac-Moody algebra assignable the Poincare-, color- and electroweak symmetries on one hand and with dynamical generated from supersymplectic symmetry assignable with the boundaries of causal diamond (CD) and extended Kac-Moody symmetres assignable to the light-like orbits of partonic 2-surfaces defining boundaries between space-time regions with Minkowskian and Euclidian signatures of the induced metric.

McKay correspondence states that the finite discrete subgroups of SU(2) can be characterized by McKay graphs characterizing the fusion rules for the tensor products for the representations of these groups. These graphs correspond to the Dynkin diagrams for Kac-Moody algebras of ADE type group (all roots have same unit length in Dynkin diagram). This inspires the conjecture that finite subgroups of SU(2) indeed correspond to Kac-Moody algebras. Could the representations of discrete subgroups appearing in the McKay graph define also representations for the ground states of corresponding ADE type Kac-Moodyt algebra? More generally, could the Mc-Kay graps of the Galois groups?

Number theoretic Langlands correspondence in turn states roughly that the representations of Galois group for extensions of rationals correspond to the so called automorphic representations of algebraic variants of reductive Lie groups. This is not totally surprising since the matrices defining algebraic matrix group has matrix elements in the extension of rationals. This raises the question how closely the number theoretic Langlands correspondence corresponds to the basic physical picture of TGD.

1. Could normal sub-groups of symplectic group and of Galois groups correspond to each other?

Measurement resolution realized in terms of various inclusion is the key principle of quantum TGD. There is an analogy between the hierarchies of Galois groups, of fractal sub-algebras of supersymplectic algebra (SSA), and of inclusions of hyperfinite factors of type II1 (HFFs). The inclusion hierarchies of isomorphic sub-algebras of SSA and of Galois groups for sequences of extensions of extensions should define hierarchies for measurement resolution. Also the inclusion hierarchies of HFFs are proposed to define hierarcies of measurement resolutions. How closely are these hierarchies related and could the notion of measurement resolution allow to gain new insights about these hierarchies and even about the mathematics needed to realize them?

1. As noticed, SSA and its isomorphic sub-algebras are in a relation analogous to the between normal sub-group H of group Gal (analog of isomorphic sub-algebra) and the group G/H. One can assign to given Galois extension a hierarchy of intermediate extensions such that one proceeds from given number field (say rationals) to its extension step by step. The Galois groups H for given extension is normal sub-group of the Galois group of its extension. Hence Gal/H is a group. The physical interpretation is following. Finite measurement resolution defined by the condition that H acts trivially on the representations of Gal implies that they are representations of Gal/H. Thus Gal/H is completely analogous to the Kac-Moody type algebra conjecture to result from the analogous pair for SSA.

2. How does this relate to McKay correspondence stating that inclusions of HFFs correspond to finite discrete sub-groups of SU(2) acting as isometries of regular n-polygons and Platonic solids correspond to Dynkin diagrams of ADE type Super Kac-Moody algebras (SKMAs) determined by ADE Lie group G. Could one identify the discrete groups as Galois groups represented geometrically as sub-groups of SU(2) and perhaps also those of corresponding Lie group? Could the representations of Galois group correspond to a sub-set of representations of G defining ground states of Kac-Moody representations. This might be possible. The sub-groups of SU(2) can however correspond only to a very small fraction of Galois groups.

Can one imagine a generalization of ADE correspondence? What would be required that the representations of Galois groups relate in some natural manner to the representations as Kac-Moody groups.

1.1 Some basic facts about Galois groups and finite groups

Some basic facts about Galois groups mus be listed before continuing. Any finite group can appear as a Galois group for an extension of some number field. It is known whether this is true for rationals (see this).

Simple groups appear as building bricks of finite groups and are rather well understood. One can even speak about periodic table for simple finite groups (see this). Finite groups can be regarded as a sub-group of permutation group Sn for some n. They can be classified to cyclic, alternating , and Lie type groups. Note that alternating group An is the subgroup of permutation group Sn that consists of even permutations. There are also 26 sporadic groups and Tits group.

Most simple finite groups are groups of Lie type that is rational sub-groups of Lie groups. Rational means ordinary rational numbers or their extension. The groups of Lie type (see this) can be characterized by the analogs of Dynkin diagrams characterizing Lie algebras. For finite groups of Lie type the McKay correspondence could generalize.

1.2 Representations of Lie groups defining Kac-Moody ground states as irreps of Galois group?

The goal is to generalize the McKay correspondence. Consider extension of rationals with Galois group Gal. The ground staes of KMA representations are irreps of the Lie group G defining KMA. Could the allow ground states for given Gal be irreps of also Gal?

This constraint would determine which group representations are possible as ground states of SKMA representations for a given Gal. The better the resolution the larger the dimensions of the allowed representations would be for given G. This would apply both to the representations of the SKMA associated with dynamical symmetries and maybe also those associated with the standard model symmetries. The idea would be quantum classical correspondence (QCC) space-time sheets as coverings would realize the ground states of SKMA representations assignable to the various SKMAs.

This option could also generalize the McKay correspondence since one can assign to finite groups of Lie type an analog of Dynkin diagram (see this). For Galois groups, which are discrete finite groups of SU(2) the hypothesis would state that the Kac-Moody algebra has same Dynkin diagram as the finite group in question.

To get some perspective one can ask what kind of algebraic extensions one can assign to ADE groups appearing in the McKay correspondence? One can get some idea about this by studying the geometry of Platonic solids (see this). Also the geometry of Dynkin diagrams telling about the geometry of root system gives some idea about the extension involved.

1. Platonic solids have p vertices and q faces. One has [p,q]∈ { [3, 3], [4, 3], [3, 4], [5, 3], [3, 5]}. Tetrahedron is self-dual (see this) object whereas cube and octahedron and also dodecahedron and icosahedron are duals of each other. From the table of Wikipedia article one finds that the cosines and sines for the angles between the vectors for the vertices of tetrahedron, cube, and octahedron are rational numbers. For icosahedron and dodecahedron the coordinates of vertices and the angle between these vectors involve Golden Mean φ=(1+51/2)/2 so that algebraic extension must involve 51/2 at least.

The dihedral angle θ between the faces of Platonic solid [p,q] is given by sin(θ/2)= cos(π/q)/sin(π/p). For tetrahedron, cube and octahedron sin(θ) and cos(θ) involve 31/2. For icosahedron dihedral angle is tan(θ/2)= φ. For instance, the geometry of tetrahedron involves both 21/2 and 31/2. For dodecahedron more complex algebraic numbers are involved.

2. The rotation matrices for for the triangles of tetrahedron and icosahedron involve cos(2π/3) and sin(2π/3) associated with the quantum phase q= exp(i2π/3) associated with it. The rotation matrices performing rotation for a pentagonal face of dodecahedron involves cos(2π/5) and sin(2π/5) and thus q= exp(i2π/5) characterizing the extension. Both q= exp(i2π/3) and q= exp(i2π/5) are thus involved with icosahedral and dodecahedral rotation matrices. The rotation matrices for cube and for octahedron have rational matrix elements.

3. The Dynkin diagrams characterize both the finite discrete groups of SU(2) and those of ADE groups. The Dynkin diagrams of Lie groups reflecting the structure of corresponding Weyl groups involve only the angles π/2, 2π/3, π-π/6, 2π- π/6 between the roots. They would naturally relate to quadratic extensions.

For ADE Lie groups the diagram tells that the roots associated with the adjoint representation are either orthogonal or have mutual angle of 2π/3 and have same length so that length ratios are equal to 1. One has sin(2π/3)= 31/2/2. This suggests that 31/2 belongs to the algebraic extension associated with ADE group always. For the non-simply laced Lie groups of type B, C, F, G the ratios of some root lengths can be 21/2 or 31/2.

For ADE groups assignable to n-polygons (n>5) Galois group must involve the cyclic extension defined by exp(i2π/n). The simplest option is that the extension corresponds to the roots of the polynomial xn= 1.

2. A possible connection with number theoretic Langlands correspondence

I have discussed number theoretic version of Langlands correspondence in \citeallb/Langland,Langlandsnew trying to understand it using physical intuition provided by TGD (the only possible approach in my case). Concerning my unashamed intrusion to the territory of real mathematicians I have only one excuse: the number theoretic vision forces me to do this.

Number theoretic Langlands correspondence relates finite-dimensional representations of Galois groups and so called automorphic representations of reductive algebraic groups defined also for adeles, which are analogous to representations of Poincare group by fields. This is kind of relationship can exist follows from the fact that Galois group has natural action in algebraic reductive group defined by the extension in question.

The "Resiprocity conjecture" of Langlands states that so called Artin L-functions assignable to finite-dimensional representations of Galois group Gal are equal to L-functions arising from so called automorphic cuspidal representations of the algebraic reductive group G. One would have correspondence between finite number of representations of Galois group and finite number of cuspidal representations of G.

This is not far from what I am naively conjecturing on physical grounds: finite-D representations of Galois group are reductions of certain representations of G or of its subgroup defining the analog of spin for the automorphic forms in G (analogous to classical fields in Minkowski space). These representations could be seen as induced representations familiar for particle physicists dealing with Poincare invariance. McKay correspondence encourages the conjecture that the allowed spin representations are irreducible also with respect to Gal. For a childishly naive physicist knowing nothing about the complexities of the real mathematics this looks like an attractive starting point hypothesis.

In TGD framework Galois group could provide a geometric representation of "spin" (maybe even spin 1/2 property) as transformations permuting the sheets of the space-time surface identifiable as Galois covering. This geometrization of number theory in terms of cognitive representations analogous to the use of algebraic groups in Galois correspondence might provide a totally new geometric insights to Langlands correpondence. One could also think that Galois group represented in this manner could combine with the dynamical Kac-Moody group emerging from SSA to form its Langlands dual.

Skeptic physicist taking mathematics as high school arithmetics might argue that algebraic counterparts of reductive Lie groups are rather academic entities. In adelic physics the situation however changes completely. Evolution corresponds to a hierarchy of extensions of rationals reflected directly in the physics of dark matter in TGD sense: that is as phases of ordinary matter with heff/h=n identifiable as order of Galois group for extension of rationals. Algebraic groups and their representations get physical meaning and also the huge generalization of their representation to adelic representations makes sense if TGD view about consciousness and cognition is accepted.

In attempts to understand what Langlands conjecture says one should understand first the rough meaning of many concepts. Consider first the Artin L-functions appearing at the number theoretic side. Consider first the Artin L-functions appearing at the number theoretic side.

1. L-functions (see this) are meromorphic functions on complex plane that can be assigned to number fields and are analogs of Riemann zeta function factorizing into products of contributions labelled by primes of the number field. The definition of L-function involves Direchlet characters: character is very general invariant of group representation defined as trace of the representation matrix invariant under conjugation of argument.

2. In particular, there are Artin L-functions (see this) assignable to the representations of non-Abelian Galois groups. One considers finite extension L/K of fields with Galois group G. The factors of Artin L-function are labelled by primes p of K. There are two cases: p is un-ramified or ramified depending on whether the number of primes of L to which p decomposes is maximal or not. The number of ramified primes is finite and in TGD framework they are excellent candidates for physical preferred p-adic primes for given extension of rationals.

These factors labelled by p analogous to the factors of Riemann zeta are identified as characteristic polynomials for a representation matrix associated with any element in a preferred conjugacy class of G. This preferred conjugacy class is known as Frobenius element Frob(p) for a given prime ideal p , whose action on given algebraic integer in OL is represented as its p:th power. For un-ramified p the characteristic polynomial is explicitly given as determinant det[I-tρ(Frob(p))]-1, where one has t= N(p)-s and N(p) is the field norm of p in the extension L (see this).

In the ramified case one must restrict the representation space to a sub-space invariant under inertia subgroup, which by definition leaves invariant integers of OL/p that is the lowest part of integers in expansion of powers of p.

At the other side of the conjecture appear representations of algebraic counterparts of reductive Lie groups and their L-functions and the two number theoretic and automorphic L-functions would be identical.
1. Automorphic form F generalizes the notion of plane wave invariant under discrete subgroup of the group of translations and satisfying Laplace equation defining Casimir operator for translation group. Automorphic representations can be seen as analogs for the modes of classical fields with given mass having spin characterized by a representation of subgroup of Lie group G (SO(3) in case of Poincare group).

Automorphic functions as field modes are eigen modes of some Casimir operators assignable to G. Algebraic groups would in TGD framework relate to adeles defined by the hierarchy of extensions of rationals (also roots of e can be considered in extensions). Galois groups have natural action in algebraic groups.

2. Automorphic form (see this) is a complex vector valued function F from topological group to some vector space V. F is an eigen function of certain Casimir operators of G. In the simplest situation these function are invariant under a discrete subgroup Γ⊂ G identifiable as the analog of the subgroup defining spin in the case of induced representations.

In general situation the automorphic form F transforms by a factor j of automorphy under Γ. The factor can also act in a finite-dimensional representation of group Γ, which would suggest that it reduces to a subgroup of Γ obtained by dividing with a normal subgroup. j satisfies 1-cocycle condition j(g1,g2g3)= j(g1g2,g3) in group cohomology guaranteeing associativity (see this). Cuspidality relates to the conditions on the growth of F at infinity.

3. Elliptic functions in complex plane characterized by two complex periods are meromorphic functions of this kind. A less trivial situation corresponds to non-compact group G=SL(2,R) and Γ ⊂ SL(2,Q).

There are more groups involved: Langlands group LF and Langlands dual group LG. A more technical formulation says that the automorphic representations of a reductive Lie group G correspond to homomorphisms from so called Langlands group LF (see this) at the number theoretic side to L-group LG or Langlands dual of algebraic G at group theory side (see this). It is important to notice that LG is a complex Lie group. Note also that homomorphism is a representation of Langlands group LF in L-group LG. In TGD this would be analogous to a homomorphism of Galois group defining it as subgroup of the group G defining Kac-Moody algebra.
1. Langlands group LF of number field is a speculative notion conjectured to be a extension of the Weil group of extension, which in turn is a modification of the absolute Galois group. Unfortunately, I was not able to really understand the Wikipedia definition of Weil group (this). If E/F is finite extension as it is now, the Weil group would be WE/F= WF/WcE, WcE refers to the commutator subgroup WE defining a normal subgroup, and the factor group is expected to be finite. This is not Galois group but should be closely related to it.

Only finite-D representations of Langlands group are allowed, which suggests that the representations are always trivial for some normal subgroup of LF For Archimedean local fields LF is Weil group, non-Archimedean local fields LF is the product of Weil group of L and of SU(2). The first guess is that SU(2) relates to quaternions. For global fields the existence of LF is still conjectural.

2. I also failed to understand the formal Wikipedia definition of the L-group LG appearing at the group theory side. For a reductive Lie group one can construct its root datum (X*,Δ,X*, Δc), where X* is the lattice of characters of a maximal torus, X* its dual, Δ the roots, and Δc the co-roots. Dual root datum is obtained by switching X* and X* and Δ and Δc. The root datum for G and LG are related by this switch.

For a reductive G the Dynkin diagram of LG is obtained from that of G by exchanging the components of type Bn with components of type Cn. For simple groups one has Bn↔ Cn. Note that for ADE groups the root data are same for G and its dual and it is the Kac-Moody counterparts of ADE groups, which appear in McKay correspondence. Could this mean that only these are allowed physically?

3. Consider now a reductive group over some field with a separable closure K (say k for rationals and K for algebraic numbers). Over K G as root datum with an action of Galois group of K/k. The full group LG is the semi-direct product LG0⋊ Gal(K/k) of connected component as Galois group and Galois group. Gal(K/k) is infinite (absolute group for rationals). This looks hopelessly complicated but it turns it that one can use the Galois group of a finite extension over which G is split. This is what gives the action of Galois group of extension (l/k) in LG having now finitely many components. The Galois group permutes the components. The action is easy to understand as automorphism on Gal elements of G.

Could TGD picture provide additional insights to Langlands duality or vice versa?
1. In TGD framework the action of Gal on algebraic group G is analogous to the action of Gal on cognitive representation at space-time level permuting the sheets of the Galois covering, whose number in the general case is the order of Gal identifiable as heff/h=n. The connected component LG0 would correspond to one sheet of the covering.

2. What I do not understand is whether LG =G condition is actually forced by physical contraints for the dynamical Kac-Moody algebra and whether it relates to the notion of measurement resolution and inclusions of HFFs.

3. The electric-magnetic duality in gauge theories suggests that gauge group action of G on electric charges corresponds in the dual phase to the action of LG on magnetic charges. In self-dual situation one would have G=LG. Intriguingly, CP2 geometry is self-dual (Kähler form is self-dual so that electric and magnetic fluxes are identical) but induced Kähler form is self-dual only at the orbits of partonic 2-surfaces if weak form of electric-magnetic duality holds true. Does this condition leads to LG=G for dynamical gauge groups? Or is it possible to distinguish between the two dynamical descriptions so that Langlands duality would correspond to electric-magnetic duality. Could this duality correspond to the proposed duality of two variants of SH: namely, the electric description provided by string world sheets and magnetic description provided by partonic 2-surfaces carrying monopole fluxes?

See the new chapter Are higher structures needed in the categorification of TGD? of "Towards M-matrix" or the article with the same title.

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