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.
Answer
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 HoĊava-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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