An Exceptionally Technical Discussion of AESToE

[Lawrence B. Crowell]

rntsai asked "how?" is "GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8)".

I might not do as good a job as Lawrence B Crowell would do,
here is my attempt at showing "how?":

The Galois field GF(4) = {0,1,w,w^2}
where
w = (1/2)( - 1 + sqrt(3) i )
and
w^2 = (1/2)( - 1 - sqrt(3) i )

Look at the 0 as the zero or origin as the central dot of the D_4 Dynkin diagram visualized as being in the complex plane centered on the origin

*
\
*--* ( I used two -- to make the length more nearly equal to that of \ and / )
/
*

and
look at the 1 as the outer Dynkin dot on the right-hand side
and
look at the w as the upper outer Dynkin dot on the left-had side
and
look at the w^2 as the lower outer Dynkin dot on the left-hand side.

Tony Smith

You got it perfectly, and in this way higher Galois fields are cyclotomic elements which are Coxeter-Dynkin diagrams for various groups. In this case it is just the nice pretty triangular pattern on the argand plane. The Galois field GF(9) gives a unit element and 8-elements which are one "oct-layer," and since GF(9) = Z(i)/3Z(i), and GF(9)xGF(9) =
Z(z)/3Z(z) there are two other "layers" which define the {3, 4, 3} polytope or 24-cell. The fun goes on from there.

Lawrence B. Crowell

Lawrence B. Crowell

 

[Kea]

I wonder if these 3x3 matrices are related to the J^3(O). This is the octonion 3x3.

This is certainly one connection we have in mind. kneemo is the local guru on Jordan algebras and matrix models (another nice part of string theory). With the higher categories, non-associativity is quite natural, but there is plenty of work to do in understanding how these algebras arise from the underlying operads.

 

[rntsai]

The paper http://arxiv.org/abs/quant-ph/9802061 appears to be doing something similar to what I posted last weekend.

This construction works with codes that contain their dual. Self dual
codes (like the Hamming [8,4,4], Golay [24,12,8],...) are obviously covered
I don't know how you constructed the classical code with e8, if it
contains its dual then this will work for it. In my opinion, this
construction is a little restricitive. A lot of good quantum codes
are not constructed this way.

 

[CarlB]

... Then the elements of the Jordan matrix would be the J and M elements JM = wMJ which are cube roots of unity. ...

There is a longer explanation for the J and M cube roots of unity, including how I found them and why, at my blog here:
http://carlbrannen.wordpress.com/2008/02/06/qutrit-mutually-unbiased-bases-mubs/

I should probably add that "J" is interpreted as the generation quantum number as the Koide mass formulas are obtained from the eigenstates of J. The eigenstates of J (in pure density matrix form) are the matrices which (a) have trace 1, (b) are idempotent: $$\rho^2 = \rho$$, and (c) are eigenstates of J (and therefore are circulant 3x3 matrices).

With the Koide mass formula, the leptons are supposed to be color singlets built from preons. The "M" operator picks out the individual preons so it is the color operator. It has three eigenstates, the diagonal pure density matrices (i.e. diagonals given by (0,0,1), (0,1,0), or (1,0,0) ). Therefore M is the operator for color, which takes eigenvalues of $$\exp(2i n \pi/3)$$ for n=0, 1, 2. These eigenvalues are the same eigenvalues of the J operator, naturally, but the J operator is interpreted here as giving the generation number.

The latest post classifies the MUBs of the Dirac algebra, writes down their (very obvious) quantum numbers, and shows what might happen when you break the symmetry by assuming that the Dirac bilinears are operators with different weights:
http://carlbrannen.wordpress.com/2008/02/07/mubs-and-symmetry-breaking/

The short form description is as follows: If we assign the following "weights" to the Dirac bilinears:

1: 1
x, y, z, t: 3
xy, yz, xz, xt, yt, zt: 9
xyz, xyt, xzt, yzt: 27
xyzt: 81

and ignore signs (taking absolute values so that, for instance, an anti-particle is given the same weight as a particle in a given basis) one finds that the sum of the weights for the four particles in the 5 mutually unbiased bases are 40, 40, 40, 40, 100.

So this sort of symmetry breaking (where one assigns a weight according to the blade), results in a symmetry breaking of the 5 MUBs of the Dirac algebra into a singlet and a quad. This is apparently true for all Dirac algebra MUBs that use the Dirac bilinears as I think I enumerated them correctly.

However, one can take the Dirac bilinears and modify them by a transformation like $$B \to SBS^{-1}$$ to get a new set of Dirac elements that satisfy the relations of the bilinears. This will change the above weight distribution. The reason is that these sorts of transformations mix which MUB gets the "xyzt" contribution which dominates the weights.

Right now I'm wondering about the MUBs for C(4,1), the generalization of the Dirac algebra by adding one (hidden) dimension. There will be 9 basis sets and each will have 8 particles. The action of symmetry breaking on this by distinguishing blades is probably going to be reasonably similar, but the action of $$B \to SBS^{-1}$$ will be more interesting as this transform maps the psuedoscalar to itself.

 

[Berlin]

Back to Lisi's E8

Gets more interesting every day here, and I understand less. If we focus again on Garretts starting point: What is essentially changes in his results? I am wondering if we could just change the E8 root numbers into complex numbers, change 0.5 to 0.5 +i*0.5*sqrt(3) etc. and connect the three generations in this way. Surely very naive. Is it established here that we need three E8's?

jan

 

[Lawrence B. Crowell]

Gets more interesting every day here, and I understand less. If we focus again on Garretts starting point: What is essentially changes in his results? I am wondering if we could just change the E8 root numbers into complex numbers, change 0.5 to 0.5 +i*0.5*sqrt(3) etc. and connect the three generations in this way. Surely very naive. Is it established here that we need three E8's?

jan

The Galois group GF(4) is the Dynkin diagram for D_4 as the cube root of unity or cyclotomic field. For the 24-cell this requires the Galois group GF(9), and the quotient Z(z)/3Z(z) defines quaternions on the 24-cell. For the E_8 the equivalent involves the octonions. I might get to that latter

To start (baby steps) a supersymmetric form of this might be needed. A start might be to consider a "naive" super field formalism for the graviton. The vierbein for gravity $(e_i)^\mu$ can be extended into a super-bein $(E_i)^\mu$ as

$$(E_i)^\mu~=~(e_i)^\mu~+~\theta^{a}\psi_{ia}^\mu~-~{\bar\theta}^{a}{\bar\psi}_{ia}^\mu~+~\theta{\bar\theta}F~+~HC$$

and the field $\psi_{ib}^\mu$ is a Rarita Schwinger bein-field. The variation of the super field is parameterized by the Grassmannian parameters $\xi,~{\bar\xi}$ as

$$\delta_{\xi}\Phi~=~({\bar\xi}_aQ^a~+~\xi_{a} {Q^a}^\dagger)\Phi$$

where Q is the supercharge operator

$$Q~=~\frac{\partial}{\partial\theta^a}~+~i{\bar\theta}_{a}\gamma^\nu\partial_\nu,$$

and the conjugate supercharge operator is easily seen. This then gives the variation of the super field

$$\delta_\xi (E_i)^\mu~=~\xi^a( \psi_{ia}^\mu~+~{\bar\theta_aF~-~i{\bar\theta}^b\theta^a(\gamma^\nu\partial_\nu\psi_{ib}^\mu~+~\partial(e_i)^\mu)$$

which defines a super-covariant differential cryptically written as $D~=~\partial/\partial\theta~+~i{\bar\theta}\gamma\cdot\partial$. The supercharge operators also in general obey anti-commutators

$$\{Q,~Q^\dagger\}~=~2i\partial~+~A$$

where the term A is a gauge potential which emerges in higher supersymmetry N > 1. These of course are of great importance, but for now I will leave that for another day.

We may of course frame this by $E~=~dx^i(E_i)^\mu\gamma_\mu$. This frame is over super-partners, and this is where we connect up with the possible pairing of E_8's. Dual to the vierbein are $\omega^\mu_{S}$ and $\omega^\mu_T$, where this duality is between the Clifford vector generators and the Clifford bivectors.

Things become from here a bit complicated. The curvature terms (ref Lisi sec 3 in particular eqn 3.3) have to be reformulated according to super-fields, and the entire SUSY construction is made for particles on one E_8 and their superpartners on the other E_8. For instance in equation 3.3 the curvature will have the term

$$D\Omega~=~d\omega ~+~\omega\omega~+~\epsilon^{abcd}\gamma^5\gamma_b\theta\cdot\partial_c\psi_d~+~\dots$$

where the "dots" include covariant terms on the RS field. I suppose that maybe I should carve out a little bit of time and delve into this program. Conceptually it is not that difficult, it will just require working through a lot of fiddle-fuddle and details.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

There is a longer explanation for the J and M cube roots of unity, including how I found them and why, at my blog here:
http://carlbrannen.wordpress.com/2008/02/06/qutrit-mutually-unbiased-bases-mubs/

I am reading up on this. There does appear to be some interesting structure here.

With the Koide mass formula, the leptons are supposed to be color singlets built from preons. The "M" operator picks out the individual preons so it is the color operator. It has three eigenstates, the diagonal pure density matrices (i.e. diagonals given by (0,0,1), (0,1,0), or (1,0,0) ). Therefore M is the operator for color, which takes eigenvalues of $$\exp(2i n \pi/3)$$ for n=0, 1, 2. These eigenvalues are the same eigenvalues of the J operator, naturally, but the J operator is interpreted here as giving the generation number.

E_6 has rank 6 and is dimension 78 under D_4 with the covering group is Z_2 and Z_3, where Z_3 leads the Haguchi-Hanson metric. The Z_2 is the automorphism group and Z_3 is the fundamental group. This has a fundamental representation of 27-dimensions, where three copies of these modulo three dimension for the Z_3 gives the 78 dimensions of E_6. These 27 dimensional representations give Jordan matrices J^3(V). I speculate that if the fundamental group here is given by another D_4, so its cyclotomic field defines the cyclic group, then this would give the 27 according to a Jordan matrix. It would also mean that the E_6 is embeded in a "superstructure" that contains two D_4's. Maybe a route to two E_8's?

Lawrence B. Crowell

 

[John G]

Gets more interesting every day here, and I understand less. If we focus again on Garretts starting point: What is essentially changes in his results? I am wondering if we could just change the E8 root numbers into complex numbers, change 0.5 to 0.5 +i*0.5*sqrt(3) etc. and connect the three generations in this way. Surely very naive. Is it established here that we need three E8's?

Garrett has mentioned looking at some different things (complex E8, Chang and Soo approach, Kaluza-Klein). The three E8s might be a separate use of E8 from the E8 describing the fundamental physics. The fundamental E8 might only have one generation and the three E8s could be used to describe up to three generations where the 2nd and 3rd are composite particles rather than fundamental ones, like making a proton from three quarks.

 

[Lawrence B. Crowell]

This construction works with codes that contain their dual. Self dual
codes (like the Hamming [8,4,4], Golay [24,12,8],...) are obviously covered
I don't know how you constructed the classical code with e8, if it
contains its dual then this will work for it. In my opinion, this
construction is a little restricitive. A lot of good quantum codes
are not constructed this way.

Self dual codes are important for quantum coding over GF(q) for a system with n vectors defines a space H and $H^*$ such that for any vector $u~\in~H$ then any vector v with $u\cdot v~=~0$ there exists a

$$H^*~=~\{v~\in~GF_n(p):v\cdot u~=~0,~\forall u~\in~H\}$$

The dual of a code space H has dimension $dim(H^*)~=~n~-~dim(H)$, and for this dimension n/2 then the code space is self dual, or the code is self dual. The Hamming [8, 4, 4] is the E_8 code!

The E_8 lattice defines a set of all possible permutations of coordinates which is the $D_8~\subset~E_8$. The permutations of 8 letters plus sign changes and the block diagonal given by of the Hadamard matrices H_4 $Diag(H_4\otimes H_4)$ is the automorphism group with order $|W(E_8)|~=~2^{14}3^55^27~=~626729600$.

This might be a way to consider the supersymmetrized version. If we regard each element of an E_8 as a multiplet the lattice has been "graded." Since $SL(2,~C)~=~SL(2,~R)\times SL(2,~R)$ defining each element of E_8 with a multiplet, is analogous to extending a group this way. My bet is that the full extension is teh Barnes-Wall lattice, and this I think might be shown by defining the set of automorphisms on $\Lambda_{16}$ similarly with the E_8 above.

Freedman proved the existence of a class of strange 4-manifolds, topologically called the E_8 manifold, These manifolds have transversals in moduli whose intersection form is the E_8 lattice. This class of manifolds are sometimes called "fake" for they are homeomorphic, but not diffeomorphic (smooth or $C^{\infty}$). For a number of physical reasons these appear to play a role in quantum gravity. The path integral

$$|\psi\rangle~=~\int_{\{g\}} \delta[g] e^{iS[g]}|\phi[g]\rangle$$

sums over states with metric configuration variables where most do not have classical meaning. Most of my arguments on this are physical (we are doing physics after all), but this is where the "third" E_8 I think comes into the picture which defines the Leech lattice $\Lambda_{24}$

The next step is to then "brain damage" this theory. There are 196560 roots in this theory, which are just too much to really work with. So the subgroup $S^3\times SL_2(7)$ is considered. This is a three sphere where every point is given by a three elements defined by a Fano plane, or elements which stabilize octonions. We might think of the S^3 as a form of bloch sphere in quantum mechanics. This has 1440 roots, which is a more reasonable system to try to work with as some system to do quantum cosmology with.

I might go into this later, but I think that quantum cosmology might go all the way up to monster groups and moonshine. If so this more exact theory may only be known on its surface, where with some $\sim~10^{50}$ elements we might never know this in detail.

Lawrence B. Crowell

 

[=CIA= h1tman]

Looks interesting! I just wish I could understand it all :(

 

[Tony Smith]

Lawrence B. Crowell said "... the "third" E_8 ... defines the Leech lattice ... There are 196560 roots in this theory ... quantum cosmology might go all the way up to monster groups and moonshine ...".

James Lepowsky said in math.QA/0706.4072 that "... the Fischer-Griess Monster M ... was constructed by Griess as a symmetry group (of order about 10^54) of a remarkable new commutative but very, very highly nonassociative, seemingly ad-hoc, algebra B of dimension 196,883 ... The Monster is the automorphism group of the smallest nontrival string theory that nature allows ... Bosonic 26-dimensional space-time ... "compactified" on 24 dimensions, using the orbifold construction ...".

My E6 string model CERN CDS preprint EXT-2004-031 on the web at
http://cdsweb.cern.ch/record/730325
is also based on orbifolding bosonic 26-dim string theory,
with strings physically interpreted as world-lines,
and with 8-dim Kaluza-Klein spacetime based on 8-dim branes with E8 structure.

In that model, a Single Cell can be described by
taking the quotient of its 24-dimensional O+, O-, Ov subspace
modulo the 24-dimensional Leech lattice,
and
its automorphism group is the largest finite sporadic group, the Monster Group, whose order is
808017424794512875886459904961710757005754368000000000
=
2^46 x 3^20 x 5^9 x 7^6 x 11^2 x 13^3 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71
or about 8 x 10^53.

If you use positronium (electron-positron bound state of the two lowest-nonzero-mass Dirac fermions) as a unit of mass Mep = 1 MeV,
then it is interesting that the product of the squares of the Planck mass
Mpl = 1.2 x 10^22 MeV
and W-boson mass Mw = 80,000 MeV
gives ( ( Mpl/Mep )( Mw/Mep) )^2 = 9 x 10^53
which is roughly the Monster order.

The Mpl part of M may be related to Aut(Leech Lattice) = double cover of Co1.
The order of Co1 is 2^21.3^9.5^4.7^2.11.13.23 or about 4 x 10^18.

The Mw part of M may be related to Aut(Golay Code) = M24.
The order of M24 is 2^10.3^3.5.7.11.23 or about 2.4 x 10^8.

If you look at the physically realistic superposition of 8 such Cells,
you get 8 copies of the Monster of total order about 6.4 x 10^54,
which is roughly the product of the Planck mass and Higgs VEV squared:
(1.22 x 10^22 )^2 x (2.5 x 10^5)^2 = 9 x 10^54

The full physics of that model can be regarded as an infinite-dimensional Affinization of the Theory of that Single Cell.

Tony Smith

 

[CarlB]

A basic problem with assuming an MUB model for elementary particles is that it implies 2 body interactions between states. If two states are in the same basis, the transition probability between them is zero, and if they're in different bases, the transition probability is 1/d. Unfortunately, the standard model is built with 3 body interactions.

The solution is to suppose that the MUB model covers preons that make up the standard model particles that are mediated at points in spacetime by bosons that are so heavy that we can't make them. Then the 2-body interactions for the MUB becomes 3-body interactions with a hidden boson that takes away the change in quantum numbers and delivers it to another preon. The result is that even though the MUB does not preserve quantum numbers (because it ignores the hidden boson), at our low energies quantum numbers are preserved.

The natural method of applying a symmetry breaking is to choose some bilinears as "high energy" so they are hidden in the observed particle set. The remaining, visible, quantum numbers would be the 8 that Garrett typed up. Then the reason that the particle interactions correspond to triplets of quantum number vectors that add to zero comes from the requirement that changes in the preon quantum numbers add to zero.

I typed up a verbose and winding description of this, with some references to the historical idea of loading the particles into matrix representations (which are similar to the MUB stuff) here:
http://carlbrannen.wordpress.com/2008/02/09/mubs-preons-and-lisis-e8-model/

Sorry for the bad blog post, but I'm not feeling great at the moment and want to get to bed.

 

[CarlB]

Okay, the above should have "1(d+1)" instead of 1/d.

And "not feeling great at the moment" turned into a gall bladder removal. While lying around with nothing to do I took the calculations to the next level and found that I can derive the Koide relations from the assumption that the leptons are made from MUB preons with d=8. This is consistent with one or two hidden dimensions, i.e. a C(4,1) or C(5,1) complex Clifford algebra. I will try to write it up today.

The (perhaps narcotic induced) short argument is that with d=8 MUBs, the transition probabilities between them are 1/9. But to write the interaction correctly, you have to take into account virtual bosons. In analogy with the calculations for vitual boson modification of photons, this introduces an amplitude of exp(2i pi/9) into the amplitude which is the off diagonal phase factor seen in Koide. I don't think the numbers or argument are exactly correct, I always end up with factors of 2 wrong in these things, but this is the general idea.

 

[Lawrence B. Crowell]

James Lepowsky said in math.QA/0706.4072 that "... the Fischer-Griess Monster M ... was constructed by Griess as a symmetry group (of order about 10^54) of a remarkable new commutative but very, very highly nonassociative, seemingly ad-hoc, algebra B of dimension 196,883 ... The Monster is the automorphism group of the smallest nontrival string theory that nature allows ... Bosonic 26-dimensional space-time ... "compactified" on 24 dimensions, using the orbifold construction ...".

The bosonic string can be thought of wrapped in 24 dimensions by eliminating the Tachyon degrees of freedom. Physically this is of course a wise move, for we really don't want them, and their removal can determine constraints on the theory, which is the orbifold.

The algebra B of dimension 196883 is from the Normalizer N of the Fischer-Griess M-group. The Normalizer N contans three subgroups or preimages N_x, N_y, N_z, which might be thought of as axes on S_3, where N permutes these elements. I have a sort of conjecture that the intersection form for fields defined on the $\Lambda_{24}$ is N, or related to N is some sequence or homology. I would have to calculate the Weyl group $W(\Lambda_{24})$, which is one of those TBD things.

I will say that in the spirit of Raphael's painting of the Academe, I tend to be more like Aristotle with his palm towards the Earth, and maybe not as much like Plato pointing to the sky. However, these things are interesting and I think that the Monster group might represent the final theory of physics and cosmology.

My E6 string model CERN CDS preprint EXT-2004-031 on the web at
http://cdsweb.cern.ch/record/730325
is also based on orbifolding bosonic 26-dim string theory,
with strings physically interpreted as world-lines,
and with 8-dim Kaluza-Klein spacetime based on 8-dim branes with E8 structure.

Your paper is similar to what I have penned down. I wrote some bits on this last weekend about E_6 and E_7.

In that model, a Single Cell can be described by
taking the quotient of its 24-dimensional O+, O-, Ov subspace
modulo the 24-dimensional Leech lattice,
and
its automorphism group is the largest finite sporadic group, the Monster Group, whose order is
808017424794512875886459904961710757005754368000000000
=
2^46 x 3^20 x 5^9 x 7^6 x 11^2 x 13^3 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71
or about 8 x 10^53.

The $O_{\pm},~O_v$ are on $S^3\times M_{24}$, which defines the Normalizer N of the Fischer-Griess Monster group. What is interesting is that this is a sort of "recherche" structure on the subgroup $S^3\times SL_2(7)$, where $SL_2(7)$ is the Hurwitz group for the Fano plane. $SL_2(7)$ is the "circle in triangle" diagram for the octonion multiplication table and the exceptional groups are "stabilizers" of the octonions. The $F_4/B_4$ defines the additional roots added to spin(8) to define F_4 and these roots define the map

$$spin(8)~\rightarrow~F_{4\setminus 36}~\rightarrow~OP^2$$

which is a property shared by E_6 and E_7. $E_6\times su(3)/(Z/3Z)$ and $E_7\times su(2)/(Z/2Z)$ are maximal subgroups of $E_8$. where both $E_7$ and $E_6$ under signature changes contains the conformal and Desitter groups.

The Mpl part of M may be related to Aut(Leech Lattice) = double cover of Co1.
The order of Co1 is 2^21.3^9.5^4.7^2.11.13.23 or about 4 x 10^18.

The Mw part of M may be related to Aut(Golay Code) = M24.
The order of M24 is 2^10.3^3.5.7.11.23 or about 2.4 x 10^8.

If you look at the physically realistic superposition of 8 such Cells,
you get 8 copies of the Monster of total order about 6.4 x 10^54,
which is roughly the product of the Planck mass and Higgs VEV squared:
(1.22 x 10^22 )^2 x (2.5 x 10^5)^2 = 9 x 10^54

The full physics of that model can be regarded as an infinite-dimensional Affinization of the Theory of that Single Cell.

Tony Smith

This leads into my insight above. For $M_{24}$ the automorphism of Mw being the Mathieu 24-group, the intersection form of a gauge theory with a frame over this group is $\Lambda_{48}$. There is a curious symmetry of lattices about "24" where the center density of $\Lambda_{48}$ is the same as $\Lambda_{24}$. The intersection form will be determined by the roots of the gauge invariant form, moduli and the roots of that space = 196560. This is "close" to the 196884(or 3) for the Normalizer, but some where an additional 324 (or 3) elements creep into the picture. I will be bugger-all if I can figure out this out, or how to get the normalizer this way.

I do think that to calculate things a truncated model is needed, where for $S^3\times SL_2(7)$ there are 1440 roots, which for three fano planes or three E_8's with 720 roots total the remaining 720 roots are a 2-1 covering (double covering) where this is a a "Bloch sphere." This is a potentially decent model where actual quantum states could be computed. I have a bit of an idea for a scheme to compute the roots of the system. Again being Aristotle with the hand to the ground at this point.

If you look at the physically realistic superposition of 8 such Cells,
you get 8 copies of the Monster of total order about 6.4 x 10^54,
which is roughly the product of the Planck mass and Higgs VEV squared:
(1.22 x 10^22 )^2 x (2.5 x 10^5)^2 = 9 x 10^54

Oof-dah, eight monster groups! Yikes things are getting a bit out of hand. I will have to think about the scaling argument. I have thought that the size of the monster group may have something to do with the number of distinct classical states which emerge in the universe. So there might be some sort of relationship between the scaling of energy and masses and the number of possible states in the universe. The $10^{53}$ is about the inflationary factor for cosmology.

Lawrence B. Crowell

 

[Tony Smith]

Lawrence B. Crowell said "... 196560 ... is "close" to the 196884 ... but some where an additional 324 ... elements creep into the picture ...".

The Leech lattice has
3x240 + 3x16x240 + 3x16x16x240 =
= 720 + 11,520 + 184,320 = 196,560 units.

The 196,560 Leech lattice units,
plus 300 = symmetric part of 24x24,
plus 24
produce the 196,884
that is the dimension of a representation space of the Monster.

Tony Smith

PS - What I mean by the symmetric part of 24x24 is to look at 24x24 as a square matrix with side 24.
It has 24x24 = 576 elements.
24x23/2 = 276 of them are above the diagonal.
24 are on the diagonal.
24x23/2 = 276 of them are below the diagonal.

If you split the 24x214 matrix into antisymmetric part + symmetric part,
then
the antisymmetric part has 2x276/2 = 276 elements
and
the symmetric part has 24 + 2x276/2 = 24 + 276 = 300 elements.

 

[Lawrence B. Crowell]

A basic problem with assuming an MUB model for elementary particles is that it implies 2 body interactions between states. If two states are in the same basis, the transition probability between them is zero, and if they're in different bases, the transition probability is 1/d. Unfortunately, the standard model is built with 3 body interactions.

I am reading some of the literature on this, such as Bengtsson's paper. As yet I don't have much to comment on this. I think that the concern does not involve two body interactions. This theory involves a space H and the dual H*, which is a standard Hilbert space construction in quantum mechanics. This might also play some role with quantum codes, in particular for [n, k, d] with a dual [n, n - k, d] we can have for the self dual classical code a [n, 0, d]. The ternary structure to this and what appears to be a triality structure in E_8, in /\_{24} and the monster appears to be some sort of recherche structure. Maybe the MUB plays a role --- who knows. I need to read a little more on this.

Lawrence B. Crowell

 

[Mark A Thomas]

The monster symmetries generates a D3/D7 quantum cosmology utilizing a gauge theory having one to one correspondence with the cosmological rolldown scalar. The cosmological inflation requires three monster groups having representation 196883 x 196883 x 196883 immediately after the Planck epoch from the supercooling transition to reheat where one copy degenerates to 196883^2/3. All of this involves product spaces of K3 x K3 where 4 dimensional volume expands introducing cosmological constant. This leaves two copies, 196883 x 196883 (two tensored N = 4 Super Yang Mills) to generate the standard model microphysics (includes SUGRA) at the end of the cosmological scalar rolldown at the end of the Electroweak epoch at 2.5 x 10^-9 s. All of this comes from 25 spatial dimensions wrapped on a circle (M^25 x S^1) of Planck radius at time t = 0.

 

[Bowles]

I want to learn about Monster group, Leech lattice, exceptional Lie groups and all that stuff.

What books do you recommend reading? Or are there only very specialized papers on these topics yet?

How did the participants of this thread learned these things?

 

[Tony Smith]

Bowles asked "... What books do you recommend reading ... about Monster group, Leech lattice, exceptional Lie groups and all that stuff ... ? ...".

For the Leech lattice:
the book "Sphere Packings, Lattices, and Groups" by Conway and Sloane
the book "From Error-Correcting Codes Through Sphere PackingsTo Simple Groups" by Thompson

For the Monster:
the book by Conway and Sloane listed above
the book "Vertex Operator Algebras and the Monster" by Frenkel, Lepowsky and Meurman
the paper by Lepowsky at http://arxiv.org/abs/0706.4072

For exceptional Lie groups:
the book "Lectures on Exceptional Lie Groups" by Adams
the book "Geometry of Lie Groups" by Rosenfeld
the book "Einstein Manifolds" by Besse (Besse is not a real person, but a pseudonym for a group of French mathematicians)
the books "Lie Groups and Lie Algebras" by Bourbaki (also a pseudonym for a group of French mathematicians)(there are 3 volumes - for Chapters 1-3, for Chapters 4-6, and for Chapters 7-9)

You can find other material by searching the web.
I learned stuff by reading the above and other stuff most of which I found on the web.

Tony Smith

 

[Berlin]

Wish you best

Carl: I wish you all the best with your gall bladder. It's a kind of anti-gall bladder now, hope you don't mind a little joke..

I deleted my last post, because it was incomplete. I also thought that we need a bigger gravity group. I have used the quantum numbers for the e.phi fields from Garrett for the third gen leptons. This much closer looks like the right lepton Q#'s and I hope to fix them in a preon like scheme. But what results now is that the w-R and w-L fields are assigned together with the e-T and e-S fields like permutations: +/-1, +/- 1 within the Q# w-t, w-s and w. In total 12 in 4 sets of three particles, rotated by J). I have 8 e-fields now, togeher with the four W-R/L fields.

Also the phi (-+/-/1/0) fields are coupled the same way with the W and B fields with the permutations +/1 within the Q# U, V and w. This also gives 8 phi fields and four W/B fields.

4 of the e-T and e-S fields, 4 of the phi (1,0,+,-)together with the twelve x(1,2).phi (RB) etc. fields would be the preons in my scheme. All gen three lepton states can be made of preon states made of e*phi. (4x4=16). The gen three quarks are preons made of phi (1,0,+, -) *phi (1,2) RB etc. (4x12=48) I will write down the whole matrix, but the numbers are OK. This would make gluons, W/B and W-R/L to be two-preon states as well. Just like you I have a factor of two wrong: too many e-T/S and phi (1,0,+,-) fields: should have 4, but I have eight. But I don't think it's a problem because you don't need them all for a base.

All these sets, together with the 2x3 gluons all rotates within each other with your J matrix...

The scheme looks beautiful, a nice guideline according to Dirac, but a dangerous according to Smolin.

all the best,
Jan

 

[Lawrence B. Crowell]

Lawrence B. Crowell said "... 196560 ... is "close" to the 196884 ... but some where an additional 324 ... elements creep into the picture ...".

The Leech lattice has
3x240 + 3x16x240 + 3x16x16x240 =
= 720 + 11,520 + 184,320 = 196,560 units.

The 196,560 Leech lattice units,
plus 300 = symmetric part of 24x24,
plus 24
produce the 196,884
that is the dimension of a representation space of the Monster.

Tony Smith

On $\Lambda_{24}$ the physics is going to be contained in the curvature two-forms

$$\Omega~=~\sum_{\{a\}}\Omega^a,$$

where the index a runs over the weight distributions $0^1 8^{759} 12^{2576} 16^{759} 24^1$ for a set of 196560. The intersection form $\Omega\wedge\Omega$ of two-forms over $\Lambda_{24}$ will define an intersection form which is invariant under an automorphisms of the Leech lattice. This invariant form is on a $\Lambda_{48}~=~\Lambda_{24}\otimes\Lambda_{24}$ is well defined according to a doubling in the sphere packing density. This automorphism on $\Lambda_{24}$, which maps this into a self dual copy of the first, just means that the transformation on one lattice determines those on the second so as to keep the intersection form invariant. This group will be the Conway group Co_1. The additional Conway groups Co_2 and Co_3 might come from the stability on the "one and two" lattice level by some condition on the complex

$$0~\rightarrow\Omega^0(ad-g)~^D\rightarrow~\Omega^1(ad-g)~^D\rightarrow~\Omega^2(ad-g)~\rightarrow~0$$

where D is the differential operator. The group g is determined by the M_24, and the "one chain" is dual to the 23-chain which gives the 24-chain on M_24 by a lamination or a "completion" on M_{23}, and then the above sequence on the 0-1-2 chains is dual to a sequence on the 24-23-22 chains or

$$M_{24}~\rightarrow~M_{23}~\rightarrow~M_{22}$$

So the system of forms is on the monad and duad subgroup. I figure in this way the Monster (Fischer-Griess) group then exists as a moduli space. This is also defined according to a self-duality condition on the 24 lattice in the 48 lattice with density doubling.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

The monster symmetries generates a D3/D7 quantum cosmology utilizing a gauge theory having one to one correspondence with the cosmological rolldown scalar. The cosmological inflation requires three monster groups having representation 196883 x 196883 x 196883 immediately after the Planck epoch from the supercooling transition to reheat where one copy degenerates to 196883^2/3. All of this involves product spaces of K3 x K3 where 4 dimensional volume expands introducing cosmological constant. This leaves two copies, 196883 x 196883 (two tensored N = 4 Super Yang Mills) to generate the standard model microphysics (includes SUGRA) at the end of the cosmological scalar rolldown at the end of the Electroweak epoch at 2.5 x 10^-9 s. All of this comes from 25 spatial dimensions wrapped on a circle (M^25 x S^1) of Planck radius at time t = 0.

If possible, could you give some references for this or reasons.

I find that some of this is getting a bit large so to speak. My idea is to build up a "petite" quantum gravity with a $S^3\times SL_2(7)$ (three octads in $\Lambda_{24}$) from the $\Lambda_{24}$. The idea being that we can understand this with respect to E_8, which we have at least some handle on. The monster group is frankly vast. With E_8 there exist 240 roots, with $\Lambda_{24}$ there are 196560, and my idea of concentrating on a subgroup will focus in on a state space that numbers 1440. The Leech lattice has a large number of elements, and a full state space description of this is frankly --- well LARGE. At 196560 these are a lot of states. What are they physically? Maybe these are the large number of states which define dark energy, or the vacuum (vacua) states define dark energy. If these vacua are inequivalent then this may also be physics of the earliest cosmology where

$$|\Psi\rangle~=~\int_{\{g\}}e^{iS[g]}|\phi[g]\rangle,$$

where the path integral sum is over untarily inequivalent vacua. My three-octad model is a "petite" version of just this, where there are two E_8 for particle states and their supersymmetric partners, a third E_8 for the intersection form on four manifolds which I think defines a Heisenberg uncertainty $\delta E_g~\simeq~|\nabla(g'~-~g)|^2$ for a coarse graining over different metric configuration variables for 240 distinct quantum states of gravity. The three E_8's define 720 states, and the additional 720 states are what I think are involved with dark matter and dark energy.

As a numerology sideline the three-octad model connects with the modularity of the Leech lattice. It is a weigth 12 modular form (function) defined by the theta function for the E_8 lattice

$$\theta_8(q)~=~1~+~240\sum_{n=1}^\infty div(n)q^{2n}$$

where this is also the Eisenstein E_4, and div(n) is a divisor. The Leech lattice being composed of three E_8s has a theta function cubic on $\theta_8(q)$ as

$$\Theta_{24}(q)~=~\theta_8(q)^3~-~720 q^2\prod_{n=1}^\infty(1~-~q^{2n})^{24}$$

where the numbers 240 and 720 appear prominantly.

We might have a situation where the moduli space of dimension 196884 defines a moduli space, which may on "blow ups" a'la Uhlenbech theorem defines a gauge space. And so our nice little Leech lattice theory turns out to be a tiny piece of this enormous theory with implies some 10^{54} states. This becomes a situation where the math appears to have accelerated beyond the physics. What does this number reflect? Are these the possible number of metric configuration variables for separate vacua in the earliest phase of quantum cosmology? This would be my suspicion. Yet to carry things this far requires considerable physical motivation. This is of course compounded if one proposes theories which are compositions of monster groups.

Lawrence B. Crowell

 

[Mark A Thomas]

The symmetries of the monster can be calculated using the Planck scale as the cutoff:

(4/a^2)(Mpl^2/me^2)[((Mpl^2/mn^2)^1/65536 -1.00)^-1]^1/2048 = 8.08017424…*10^53

a = fine structure constant, mn = neutron mass, Mpl = Planck mass
The neutron provides for the appropriate (quark-gluon) thermal gauge fields.

The third product term contains an electroweak gauged 4D black hole:

((Mpl^2/mn^2)^1/65536 -1.00)^-1 = (((emn)^2 SbhG/2hc)^1/65536 -1.00)^-1

Where the Bekenstein Hawking entropy Sbh is that of a 4D Schwarzschild mass: Mn = hcMpl/epiGmn^2 = 2.7048 * 10^33 g Also: Sbh = Ss = piMn^2/Mpl^2
From this one can obtain N amount of color as the number of matter fields as binary operators of a CFT as a gauge theory on the S^4.
Mn/2mn = 8.07 *10^56
The Large N black hole contains at an electroweak VEV the hidden survivable moduli space of the (one copy) monster and its unitary evaporation suggests that the Einstein Hilbert action and monster are tandem in the microphysics all the way to the end for other hotter VEV. Determination of the electroweak VEV at 76.77 GEV in the theory determines the scaling properties of the gauge theory coming down from Planckian energy using the field theoretic trace value: <phi> = 744 at the Planck energy and changing KK modes associated with momenta. (generically, Mpl/744 = 1.64 *10^16 GeV)
At the end of the Plank epoch at 10^-44 seconds the entropy is an orderly: S = pi*22*196883^3 determined from the running gauge.
Between the super cooling and reheat transition two copies survive while one is broken. The growth of 4D is due to K3xK3 here and the 8.07 *10^56 matter fields are designated as vacua since ordinary matter is yet to exist. The broken copy does not retain as a Hilbert space due to distortion caused by Planckian densities and selection process (in Planck units): 196883^3 x 22 > 196883 x 22^3 [8.07*10^56 x 8.07*10^56] = 1.37 *10^123 vacuum density At reheat (GUT energy 1.64 *10^16 GeV) this vacuum density calculates to the absurd canceling QFT value: 196883 x 22^3/(196883^1/3)(22) x [8.07 *10^56 x 8.07*10^56] = 1.067 *10^120 (Planck units)
So I am looking at representation space of the monster to contain a large unitary moduli space so that two copies coming off the cosmology produce a supersymmetric tensored space of two N = 4 Super Yang Mills theories that is dual to the frame work of a 4D gauge theory that explains a one to one correspondence between the cosmological scalar roll down from initial conditions to the black hole history of a AdS5 x S^4 in evaporative time. When the path integral is taken for the cosmology and the black hole history the K3 metrics is a natural candidate with lots of room with what you want to do with compactification or product spaces. It’s just a step to go to CYs.
As to the question of how large is too large in the early cosmology I guess it depends what you are looking at or for. At this stage I am more involved with the dynamics than how the three generations are made specific. If this was all confusing you can go to my web page for more information: http://monstrousgaugetheory.googlepages.com/home

 

[MTd2]

the book "Sphere Packings, Lattices, and Groups" by Conway and Sloane

I would ask the same thing as the other guy, but he came up with the question before. This book is indeed highly recommended, but it is not available anywhere. If you can help me and him to find it, I would be really thankful. :)

 

[Emanuel]

Does the publisher ship to where you live?

 

[MTd2]

Yes, sure. But it is not available there "Order now, and we will ship *when available*. (Your credit card will not be charged until we ship)"

 

[patfla]

I found it (the book) on books.google.com

Click on 'Google Product Search' on the right at the link below. I got 6 six hits. Not cheap, but well, there you are (anywhere you go - there you are [Kurt Vonnegut]):

http://books.google.com/books?id=upYwZ6cQumoC

 

[Tony Smith]

I just went to the amazon.com page for the Conway and Sloane book
Sphere Packings, Lattices, and Groups
where I did see "Currently unavailable"
but
I also saw that I could for $19.80 order the book in digital form,
which let's you read the entire book by logging into amazon
so that you can not only read the whole thing on your computer,
you can copy and print whatever you want from the book.

Maybe this is the beginning of the end for paper books,
and
years from now people will wonder why we had paper libraries,
because we can either log into the web to read our books
or carry them all around in iPod memory.

Tony Smith

PS - Another monster book is
Moonshine beyond the Monster:
The Bridge Connecting Algebra, Modular Forms and Physics
by Terry Gannon

 

[MTd2]

I also saw that I could for $19.80 order the book in digital form,

In what format does it come?

 

[patfla]

Yes. I can't find (at Amazon) this 19.80 thing you refer to Tony. Could you be more specific?

 

[Lawrence B. Crowell]

The symmetries of the monster can be calculated using the Planck scale as the cutoff:

(4/a^2)(Mpl^2/me^2)[((Mpl^2/mn^2)^1/65536 -1.00)^-1]^1/2048 = 8.08017424…*10^53

a = fine structure constant, mn = neutron mass, Mpl = Planck mass
The neutron provides for the appropriate (quark-gluon) thermal gauge fields.

The third product term contains an electroweak gauged 4D black hole:

((Mpl^2/mn^2)^1/65536 -1.00)^-1 = (((emn)^2 SbhG/2hc)^1/65536 -1.00)^-1

Where the Bekenstein Hawking entropy Sbh is that of a 4D Schwarzschild mass: Mn = hcMpl/epiGmn^2 = 2.7048 * 10^33 g Also: Sbh = Ss = piMn^2/Mpl^2
From this one can obtain N amount of color as the number of matter fields as binary operators of a CFT as a gauge theory on the S^4.
Mn/2mn = 8.07 *10^56 http://monstrousgaugetheory.googlepages.com/home

You seem to be making a scaling argument similar to Tony Smith's. Given the Monster has 196884 dimensions the Griess B-algebra in one dimension lower might be though as as defining a flux. We might think of the B-algebra as "wrapping" the one less dimension, analagous to a string on an orbifold. From there potentials for otherwise massless fields are generated. The flux due to the form penetrating the 196883 dim space (the B-algebra) then stabilizes the moduli, here the moduli being given by the B-algebra. Then by proceeding this way a tower of KK states can be generated with masses that increase up to the Planck mass.

I will need to try to follow your argument a bit better. This looks pretty much like a scaling argument, similar to what Tony Smith advanced last week.

Lawrence B. Crowell

 

[Tony Smith]

As to Amazon digital books, see

https://www.amazon.com/gp/digital/sitb/help/learn.html/ref=amb_link_3912402_1?ie=UTF8&navbar=1&details=1&pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-2&pf_rd_r=1VEXNT04VN5RCQKNXTX9&pf_rd_t=101&pf_rd_p=257590701&pf_rd_i=293522011&tag=pfamazon01-20

or if that long URL does not work
then
go to Amazon.com
then to books
then in the column on the left go to Special Formats and then to Amazon Upgrade

and you can then see how to buy the digital version of the book.
You may have to buy the paper copy first,
but even if the book is "Currently unavailable" you may be able to buy it by pre-order and then buy the digital version (viewable immediately) for an extra $19.80.

I must have purchased a copy of Conway and Sloane from Amazon, which allowed me to buy the digital version for an additional $19.80 as an upgrade,
and now I can read the book over the net.

Tony Smith

 

[MTd2]

I must have purchased a copy of Conway and Sloane from Amazon, which allowed me to buy the digital version for an additional $19.80 as an upgrade,
and now I can read the book over the net.

Tony Smith

So, this is like buying the book plus an online only access. If my internet is down, I can't see it. I guess I will just have to wait for a printed copy.

 

[Mark A Thomas]

Monstrous 4D Gauge Theory

This looks pretty much like a scaling argument,...

Lawrence B. Crowell

It is not only scaling, it is the running of the RG with a well defined gravi-scalar <phi> with increasing momenta based on a KK tower of excitations(quasi-stable = small change). The equation of monster symmetry has embedded the electroweak VEV baseline and the base gravi-scalar. All gauge couplings including gravitation are in sync. There is a Bose-Einstein distribution form in the equation whereby the black body curve can be obtained and it is wonderfully in-line with the KK distributions. A very real physics object (the black body curve) is generated and a total of 7 QFTs are obtained (from electroweak to Planck) with a 2.136 *10^14 range Higgs sector. When one looks at the gauge coupling scaling starting at electroweak, the weak form of gravity is apparent and it is scaled as the dimensionless form: 2piGmn^2/hc = 5.92*10^-39
Again where mn is the neutron mass providing the massless modes of the chiral fields (gauge fireball, glueballs...) in minkowski spacetime.

 

[Lawrence B. Crowell]

It is not only scaling, it is the running of the RG with a well defined gravi-scalar <phi> with increasing momenta based on a KK tower of excitations(quasi-stable = small change). The equation of monster symmetry has embedded the electroweak VEV baseline and the base gravi-scalar. All gauge couplings including gravitation are in sync. There is a Bose-Einstein distribution form in the equation whereby the black body curve can be obtained and it is wonderfully in-line with the KK distributions. A very real physics object (the black body curve) is generated and a total of 7 QFTs are obtained (from electroweak to Planck) with a 2.136 *10^14 range Higgs sector. When one looks at the gauge coupling scaling starting at electroweak, the weak form of gravity is apparent and it is scaled as the dimensionless form: 2piGmn^2/hc = 5.92*10^-39
Again where mn is the neutron mass providing the massless modes of the chiral fields (gauge fireball, glueballs...) in minkowski spacetime.

To be honest one physical motivation for looking at lattices as a way of doing quantum gravity & cosmology was the prospect that physics could be reduced to formalism seen in solid state physics. A lattice defines Voronoi cells which in physics are called Brillouin zones, where phonon states are computed along with the Fermi surface for the conduction band electrons. The symmetry of the lattice determines the spectra of phonons in much the same way that a symmetry group in particle physics determines the structure or states of elementary particles. The particle states are given by eigenstates of Bloch waves on a lattice, which in lattice QCD are analogously seen in Mantin periodic Lagrangians.

There is also a nice thing thing about working in this vein, for it makes the underlying basis, frame or set of states of the theory is linear. Just as we can work with solid state physics with some comparative ease, at least with weakly interacting phonons and electrons, in this light maybe the underlying theory of supergravity has a similar simple structure

So I am going to lay out a physical prescription here for how I think this is going to work. To start we consider an N dimensional space that includes spacetime, so N > 4. We then assume that a curl-like condition determines the fields on a vector $U^a$ for $U^a~=~(U^\mu,U^j)$ for j > 4 ... N. This gives a Lagrangian

$$S~=~\int d^Nx\sqrt{g}\Big(-\frac{1}{4}(\nabla_aU_b~-~\nabla_bU_a)(\nabla^aU^b~-~\nabla^bU^a)~-~\lambda(U_aU^a~-~U^2)~+~{\cal L}_{int}\Big)$$

where $\lambda$ is a Lagrange multiplier constraining the length of the N-vector.This lattice can be of various forms, in particular for a Lie group with a lattice representation. The E_8 lattice is a discrete subgroup $\Lambda_8$ of R^8 of full rank that spans R^8. This lattice is given explicitly by a discrete set of points in R^8 such that the coordinates are integers or half-integers, and the sum of the eight coordinates is an even integer. If small spheres are assigned to these points the lattice is a body centered cubic lattice (bcc), where the bcc in three dimensions is the crystalline lattice of silicon. Symbolically the lattice is,

$$\Lambda_8~=~\{x_i~\in~Z_8~\cup~(Z_8~+~1/2)_8:~\sum_ix_i~=~0~mod~2\}$$

Clearly the sum of two lattice points is another lattice point.
Assign $\phi_i$ as the field that connects gauge coefficients with the group${\cal G}$ those with ${\cal G}'$ at the $i^{th}$ side and $\psi_{i,i+1}$ as the field attaching ${\cal G}'$ at the $i^{th}$ node to the ${\cal G}$ at the $i+1^{th}$ node. The S matrix is then defined as

$$S_{i,i+1}~=~g_s\langle~|\phi_i\psi_{i,i+1}|~\rangle.$$

A local gauge transition on this matrix is then determined by the ${\cal G}'$ groups at the vertices of the edge link by $g_i^{-1}S_{i,i+1}g_{i+1}$ and $S_{i,i+1}$ is an $m\times m$ matrix of bosons. These bosons are then "link variables" for the chain. The distinction between the two groups I discuss below. When the gauge coupling $g_s$ becomes large there is a confinement process that defines a mass, which by necessity breaks any chiral symmetry. The renormalization cut offs for confinement are set by the two groups defined as $\Lambda_n$ and $\Lambda_m$, where free fermions and their gauge bosons (e.g. quarks and gluons) are free from confinement for $E~>>~\Lambda_n,~\Lambda_m$. Under this situation, where the strength of the $\cal G$ is small, the differential of the scattering matrix in a nonlinear sigma model is,

$$D_\mu S_{i,i+1}~=~\partial_\mu S_{i,i+1}~-~igA_{\mu i}S_{i,i+1}~+~igS_{i,i+1}A_{\mu i+1},$$

where the effective Lagrangian for the field theory is

$${\cal L}_{eff}~=~-\frac{1}{2g^2}\sum_i F_{ab i}{F^{ab}}_i~+~g^2\sum_i Tr|{\cal D}_\mu S_{i,i+1}|^2.$$

This is the Lagrangian for a N - 4 dimensional ${\cal G}'$ theory, where the additional dimension has been placed on the N-polygon. The last term in the Lagrangian determines a mass Lagrangian of the form

$${\cal L}_{mass}~\sim~g_s^2\sum_i(A_i~-~A_{i+1})^2.$$

The second term in the effective Lagrangian couples the vector $U^a$ to the YM field and so we write ${\cal L}_{eff}$ as

$${\cal L}_{eff}~=~-\frac{1}{4}\sum_i F_{ab i}{F^{ab}}_i~+~\frac{1}{2m^2}U^aU^b g^{cd}F_{ab}F_{bd}$$

The equations of motion are

$$\nabla_aF^{ab}~=~\frac{1}{m^2}(U_cU^b\nabla_aF^{ca}~-~U_cU^a\nabla_aF^{cb}),$$

which when decomposed into spacetime parts $\mu~=~\{1,~\dots,~4\}$ and i > 4 are

$$\partial_\mu F^{\mu i}~=~0,$$
$$\partial_\mu F^{\mu\nu}~=~-(1~+~\frac{U^2}{m^2})\partial_iF^{i\nu}$$

We chose the gauge $A_i~=~0$ and the DEs of motion then indicate that $k_\mu k^{\mu}~=~(1~+~(U/m)^2)k_ik^i$. If we put in a mass term in the Lagrangian, such as the one implied above and equate $M^2~=~k_\mu k^\mu$ we then have

$$M^2~=~m_0^2~+~(1~+~(U/m)^2)(n^2\hbar/R)^2,$$

where the compactified dimension on i are expressed according to the compactified radius and the winding number n. In this way the mass of the gauge particle (analogous to a massive phonon) is renormalized in much the same way massive particles have renormalized masses in a Brillouin zones. This is one way to explicitely construct towers of masses.

If you look at Chapter 24 in Conway & Sloane this discusses the twenty three constructions of the Leech lattice. There are 23 Niemeier construction of the Leech Lattice. For a flat 24-dimensional space one choice works well enough. However, in general this lattice may be deformed or defined on a curved manifold. Therefore, without belaboring the point too much, there will by homology considerations be "defects" in any tesselation of the 24-dimensional manifold. The particular vectors, say the $U^a$ above will have a particular gluing, but in general an element might be connected to another with a different gluing. This is the meaning of the different groups ${\cal G}$ and ${\cal G}'$ for distinct "glue codes" in the A-D-E classification.

Lawrence B. Crowell