Mark A Thomas said:
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
<br />
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)<br />
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,
<br />
\Lambda_8~=~\{x_i~\in~Z_8~\cup~(Z_8~+~1/2)_8:~\sum_ix_i~=~0~mod~2\} <br />
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
<br />
S_{i,i+1}~=~g_s\langle~|\phi_i\psi_{i,i+1}|~\rangle.<br />
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,
<br />
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},<br />
where the effective Lagrangian for the field theory is
<br />
{\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.<br />
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
<br />
{\cal L}_{mass}~\sim~g_s^2\sum_i(A_i~-~A_{i+1})^2.<br />
The second term in the effective Lagrangian couples the vector U^a to the YM field and so we write {\cal L}_{eff} as
<br />
{\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}<br />
The equations of motion are
<br />
\nabla_aF^{ab}~=~\frac{1}{m^2}(U_cU^b\nabla_aF^{ca}~-~U_cU^a\nabla_aF^{cb}),<br />
which when decomposed into spacetime parts \mu~=~\{1,~\dots,~4\} and i > 4 are
<br />
\partial_\mu F^{\mu i}~=~0,<br />
<br />
\partial_\mu F^{\mu\nu}~=~-(1~+~\frac{U^2}{m^2})\partial_iF^{i\nu}<br />
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
<br />
M^2~=~m_0^2~+~(1~+~(U/m)^2)(n^2\hbar/R)^2,<br />
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
