It might be that the ultimate structure of the universe is monstrous-moonshine. The potential role is with the interior of that $\hbar,~i$ planar region with the various string theories connected by a region which is not well known. If the $27$ dimensional system is the Jordan matrix algebra $J^3({\cal O})$, there are three scalar elements on the diagonal and the $E_8$, or octonions, ${\cal O}_i,~i~-~1,~2,~3$. The infinite momentum constraint on the diagonals, with metric signature $[-,+,+]$ reduces this to $26$ dimensions.
The following paper:
Borcherds symmetries in M-theory
Dedicated to Pr. S. Hawking on his 60th birthday.
Pierre Henry-Labord`ere, Bernard Julia and Louis Paulot
http://arxiv.org/PS_cache/hep-th/pdf/0203/0203070v2.pdf
should be of interest.
The connections with $n~=~2,~3,~4$ n-qubit systems is with the automorphism groups of the groups, or in $24$ dimensions $Aut(M_{24}),~{\mathbb Z}_n~=~\Lambda_{24}$, which is the Leech lattice. The bipartite, tri-partite (qutrite) and 4-qubit entanglements are associated with the particular sublattice systems, $D_4$, $E_8$ and $\Lambda_{16}$, where the last is the Barnes-Wall lattice. The above infinite momentum constraint on the $3$ scalars reduces this to symmetries on a $2$ dimensional hyperbolic space with the two dimensional group, $SL(2,R)$, which are the isometries of the $AdS_2$. Without the constraint this is the isometries of $AdS_3$. This then does segue into Witten’s three dimensional gravity.
What Lobos indicates with this being a bosonic theory is worthy of note. However, with string theory we start with the bosonic string in $26$ dimensions, which is the dimension where the Kac-Moody anomaly cancellation occurs. The structures above have strong connections with $26$ dimensional string. The dimensions of $24$, $26$ and $27$ are particularly interesting. $24$ dimensions is the dimension of the Leech lattice, $26$ is half of the $F_4$ group, $dim(F_4)~=~52$
The discrete group $PSL(2,~7)$ is the automorphism group of the octonionic Fano plane. For complex-octonions the automorphism is $F_4$ of 52 dimensions. $F_4$ is equivalent to adding 16 short roots vectors to the four roots of the group $SO(9)$. This is also the symmetry of the 24-cell. The short roots define the quotient $F_4/SO(9)$ which defines the sequence:
$$
F_4/SO(9):~1~\rightarrow~spin(9)~\rightarrow~F_{4\setminus 52}~\rightarrow~{\cal O}P^2,
$$
where $4\setminus 52$ means the group is restricted to $36$ of the $52$ dimensions. The ${\cal O}P^2$ defines the Cayley plane. $F_4$ is the automorphism group of the Jordan algebra with the symmetric product $X\odot Y~=~(1/2)(AY~+~YX)$. The group $F_4$ acts transitively on ${\cal O}P^2$ with isotropy subgroup $spin(9)$. For the octo-octonions the group is the exceptional $E_8\times E_8$ with heterotic structure.
This does ultimately connect to supergravity. It does so in much the same way that the bosonic string connects with supersymmetric string theory. The $F_4/B_4$ gadget fits into this. Suppose we have that $SO(16)~\rightarrow~~SO(14,2)$. Now let us restrict this to $SO(14,1)$ the Lorentz spacetime. We then have the further reduction due to the lightcone gauge or the “infinite momentum frame. This reduces dimensions further, yet we may not want to do that here. We have the $SO(16)\times SO(10)$ in the type II theory corresponding to the 26-dimensional bosonic string. This embeds into the larger $SO(26)$. So let us do our reductions on $SO(24,2)$, to $SO(24,1)$, then the infinite momentum or light cone gauge reduction is to SO(23,1).
The $SO(24)$ group (here Euclideanized for simplicity) is the string theory of quantum gravity. The $E_8\times E_8$ corresponds to the left and right moving modes on the string. Consider a heterotic string with $26$ dimension for left moving modes, which correspond to $10$ right moving modes. The k-metric is then $k^2~=~k_l^2~–~k_r^2$, which for $10$ of the $k_l$ modes $k_l~=~k_r$ component by component $1$ to $10$, then the k-metric is $16$ dimensional. This is the origin of the $E_8\times E_8$ or equivalently $SO(2*dim(k))~=~SO(32)$. The bosonic portion in $26$ dimensions contains two dimensions which are tachyonic vacuum modes and the remaining an $SO(24)$ field with tensor $\Omega_{ab}$. The string operators $\alpha^a_{-1}$ projects the vacuum $|0\rangle$ to $|\Omega_{ab}\rangle$ by
$$
\alpha^a_{-1}\alpha^b_{-1}|0\rangle~=~|\Omega^{ab}\rangle
$$
This field has a traceless portion which is the spin-2 graviton, the trace term a dilaton operator and the anti-commuting portion define a gauge field theory in the adjoint representation. This is a decomposition into $SO(8)$ and $SO(16)$, where the $SO(8)$ portion is conformal gravity and the $SO(16)$ gives the roots of $E_8$.
The $AdS_n$, $n~=~2,~3$ gravity then has a correlation with the standard $SO(24)$ bosonic string gravity in Green, Schwarz & Witten. I think there is a great prospect for the future of this, which should lead to a monstrous-moonshine realization of the universe.
This post imported from StackExchange Physics at 2015-10-01 08:28 (UTC), posted by SE-user Lawrence B. Crowell
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