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The heterotic string is a combination of right-moving excitations from a D=10 superstring and left-moving excitations from a D=26 bosonic string, with the left-movers behaving as if the extra 16 dimensions are compactified. The heterotic string is also derived from D=11 M-theory, as an open 2-brane stretched between two "end-of-the-world" 9-branes (spatial boundaries; this is M-theory compactified on a line segment, and the 9-branes lie at the ends). So I am led to imagine a 27-dimensional theory, containing branes. We compactify 16 dimensions, and consider the worldvolume theory of two parallel 9-branes. When they are coincident, we get the heterotic string; when they are slightly separate, we get "heterotic M-theory".

A 27-dimensional fundamental theory has been discussed before (hep-th/9704158 section 4; hep-th/0012037; arXiV:0807.4899), but I don't see this particular line of thought discussed.

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I don't quite see what your question is... –  David Z Jan 24 '11 at 7:32
@David, neither do I - why wasn't it closed? –  John McVirgo Feb 8 '11 at 16:11
@John: well, there is the seed of a question here. Mitchell seems to be wondering whether there's a reason this idea is not apparently not discussed in the literature - I was just hoping to get that clarified. In any case, I do note that nobody else seems to object to the question, judging by the lack of downvotes or close votes, which suggests that if I'd closed it there would have been complaints on meta. –  David Z Feb 8 '11 at 17:47
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up vote 6 down vote accepted

Dear Mitchell, this is a very nice research project - at least judging by the fact that I have made a similar proposal. ;-)

In this very form, however, it can't be right because any hypothetical 27-dimensional theory fails to be supersymmetric and the supersymmetry breaking can't be quite undone. However, brave souls have played with the transmutation of string theories that are very different on the world sheet, see e.g. some of the papers by Simeon Hellerman and Ian Swanson:


In fact, my specific version of the proposal had one mathematical piece of evidence that was much more specific than yours. You could imagine some $E_8$ group already in 27 dimensions. And a funny feature of the $E_8$ are its holonomy groups. The nontrivial one is the $\pi_3$ which is $Z$, and then the next nontrivial one is $\pi_{15}$. In normal M-theory, with the 3-form described following Diaconescu-Moore-Witten as the Chern-Simons form of an $E_8$ gauge field, $\pi_{3}$ is what allows fivebranes (codimension 5) to exist.

Similarly, $\pi_{15}$ of $E_8$ may create codimension 17 objects, and 27-17 = 10 which is the spacetime dimension of the Hořava-Witten domain wall. Very natural. So I would actually propose you modify the proposal so that $E_8$ already exists in the bulk of 27 dimensions and you create a variation of the DMW paper at the same moment.

Otherwise, you will face a lot of trouble. The quantities are unstable, unprotected by supersymmetry, so even if the instabilities can be survived, you won't be able to match the precise numbers on both sides of a duality.

Moreover, non-fermionic theories don't carry any gauginos and they have no anomalies, so you will be able to show no nontrivial anomaly cancellation that would be similar to the anomaly cancellation of heterotic M-theory, and so on. It is simply very hard to make a convincing story of a 27-dimensional origin of the heterotic string.

Note that even the ordinary bosonic M-theory remains highly inconclusive. So far, we have only presented some analogous construction for another string vacuum - an appendix to the papers you mentioned that are not terribly important (or famous) at this moment themselves.

Best wishes Lubos

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These researches are interesting. This connects with the projective Cayley plane, which is important in the Jordan algebra. Some of the structure discussed in the paper Hisham Sati


makes contact with the Jordan algebra. A constraint on the diagonal scalars of the Jordan matrix, which is similar to a light cone or infinite momentum condition reduces the system to $26$ dimensions. Decomposed as $10~+~16$ appears to connect with a heterotic string in $10$ dimension.

The exceptional cubic matrix is an extension of the same spin structure by the isomorphism $$ J^3({\cal O})~\simeq~ R\oplus J^2({\cal O})\oplus{\cal O}. $$ The extension to the cubic matrix model with $V~\in~J^2({\cal O}$, involve $z_0~\in~R$, and $\theta~\in~{\cal O}^2$ for $V,~\theta$ vector and spinor elements in the $10~=~9~+~1$ dimensional spacetime. The spinor representation of $so(9)$ when restricted to $so(8)$ splits as ${\bf 8}_1\oplus{\bf 8}_2$, so that $so(9)~\simeq~{\cal O}^2$. The vector portion is given by the spin factor $J^2({\cal O})$ which splits according to $J^2({\cal O})~\simeq~({\bf 8}_0\oplus R)\oplus R$ in a representation of $10$ dimensional Lorentzian spacetime. Hence the spatial coordinates $V~\in~{\bf 8}\oplus R$ are associated with their superpartners $\theta~\in~{\cal O}^2$ . The three $\bf 8$ representations of $so(8)$ are mixed according to the triality operation in the cubic matrix $$ \left(\matrix{z_1 & {\cal O}_0 & {\bar{\cal O}}_2\cr {\bar{\cal O}}_0 & z_2 & {\cal O}_1\cr {\cal O}_2 & {\bar{\cal O}}_1 & z_0}\right) $$ The three $\cal O$s give $24$ degrees of freedom, which in addition to the 3 $z$'s gives a total of $27$ degrees of freedom. The exceptional group $G_2$ is the automorphism on $\cal O$, or equivalently that $F_4\times G_2$ defines a centralizer on $E_8$. The fibration $G_2~\rightarrow~S^7$ is completed with $SO(8)$, where the three ${\cal O}$'s satisfy the triality condition in $SO(8)$.

The $so(9)$ is not the most general symmetry of $J^3(0)$. There exists a permutation on the three scalars $z_0,~z_1,z_2$ and ${\cal O}^3$. This means there is an additional automorphism $so(3)$. The more general automorphism is then $F_4$. The quotient between the $52$ dimensional $F_4$ and the $36$ dimensional $so(9)~\simeq~B_4$ defines the short exact sequence $$ F_4/B_4:1~\rightarrow~spin(9)~\rightarrow~F_{4\setminus 16}~\rightarrow~{\cal O}P^2~\rightarrow~1, $$ where $F_{4\setminus 16}$ means $F_4$ restricted to $36$ dimensions, which are the kernel of the map to the $16$ dimensional Moufang or Cayley plane ${\cal O}P^2$. Geometrically the $F_4$ define the symmetry of the $24$cell, called the icositetrachoron or polyoctahedron, according to $24$ octahedral cells. The $B_4$ also defines a more restricted symmetry on the $24$ cell according to $16$ tetrahedral cells and $8$ octahedral cells. The $8$ octahedral cells define the ${\bf 8}_0$, or $so(8)$ in the $J^2({\cal O})$, while the $16$ tetrahedral cells are mapped to the ${\cal O}P^2$. This means on the algebraic level $f_4~\simeq~ so(8)\oplus V\oplus \theta_1\oplus \theta_2$, which explicitly describes the triality condition the three octonions with the $so(8)$. More generally according to octonions $f_4~\simeq~so({\cal O})\oplus {\cal O}^3$. $f_4$ diagonalizes then the Jordan cubic matrix.

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