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stub for Hořava-Witten theory
briefly added the argument on the boundary conditions for the sugra C-field to Hořava-Witten theory. (not well written, maybe I’ll polish this later. )
added to Horava-Witten theory pointer to the recent
It just occurs to me that there is also a first-principles derivation of Hořava-Witten theory from just the brane bouquet and the parameterized WZW terms that it induces:
Way back in D’Auria-Fre 81, there is this curious observation, in equation (3.14), that besides the $2+2$ cocycle for the M2-brane, there is also what would be the $10+2$ cocycle for an M10-brane on 11-dimensional super-Minkowski spacetime. This was not considered elsewhere in the brane scan.
Now there is a reason for this – but not an entirely good reason. The reason is that on 11d super-Minkowski spacetime, this 10+2 cocycle is exact. It’s (left invariant) potential is simply the 11-dimensional volume form.
This means that there is no non-trivial M10-brane on 11-dimensional super-Minkowski spacetime. But it also means this: on non-orientable 11-dimensional super-spacetimes, there is a non-trivial M10 brane. (Not by speculation, but by mathematics.)
Even better: even if the 11-dimensional superspacetime happens to be orientable, we may force a non-trivial M10-brane to appear by orientifolding it: hence by, in particular, passing to the orbifold quotient by an orientation non-preserving $\mathbb{Z}_2$-action.
Globalizing the 10+2 cocycle over an 11d orientifold hence gives the following: there is an M10 brane on such an orientifold whose supercharge is trivial almost everywhere, except rigth there are the $\mathbb{Z}_2$ fixed point loci. So charge-wise it looks like it decays to an M9-brane sitting at these fixed points.
And that of course if just what Hořava-Witten theory is.
Presumably $X_{10} \times /(S^1//\mathbb{Z}_2)$ should be $X_{10} \times (S^1//\mathbb{Z}_2)$.
Expanded the Idea-section slightly to include also the type I case, hence the picture of the triple of dualities
$\array{ HE &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}& M &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}& I' \\ \mathllap{T}\updownarrow && && \updownarrow \mathrlap{T} \\ HO && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} && I }$added disambiguation line to the entry
This entry is about the conjectured relation between M-theory at MO9-planes and heterotic string theory on these. For the relation of M-theory KK-compactified on a K3-surface and heterotic string theory on a 3-torus see instead at duality between M/F-theory and heterotic string theory
(since “duality between M-theory and heterotic string theory” redirects to here, while it could equally well redirect to there)
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