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It seems to be generally accepted ([1], [2]) that one can apply the duality between a $T^3$ compactification of heterotic string theory and a $\mathrm{K3}$ compactification of M-theory "fiberwise" to dualize a heterotic compactification on a Calabi-Yau manifold $W$ that is fibered as $T^3 \to W \to Q$ into an M-theoretic compactification on a $G_2$ manifold $X$ that is fibered as $\mathrm{K3}\to X\to Q$ and vice versa.

The question is: Given $X$, what is $W$?

[2] seems to say that we should pick a five-brane world-volume $\Sigma\times C\subset X$ with $\Sigma$ a Riemann surface and $C$ a coassociative 4-cycle, and that $W$ is given by the moduli space of deformations of $C$, called $\mathcal{M}_\text{coassoc}$, for more on which we are referred to [3]. Maybe it's also just meant to be the basis of a fibration, [2] seems seriously unclear on this point.

In any case, [3] tells us that the tangent space of $\mathcal{M}_\text{coassoc}$ is $H^2_+(C)\oplus H^1(C,\mathrm{ad}(E))$, where $E$ is a bundle with anti-self-dual connection reflecting the physical gauge fields living on the brane wrapped around the cycle. In other words, this space is most decidedly not always six- or seven-dimensional, and might even be much larger as far I can see. How is this supposed to relate to the heterotic $W$?


References

[1] : Acharya and Witten, Chiral Fermions from Manifolds of $G_2$ Holonomy, hep-th/0109152

[2] : Gukov, Yau, Zaslow, Duality and Fibrations on $G_2$ Manifolds, hep-th/0203217v1

[3] : Lee and Leung, Geometric structures on G2 and Spin(7)-manifolds, hep-th/0202045

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  • $\begingroup$ I don't know much about fibrewise dualities, but I did discuss some of these issues with Arthur Hebecker (in relation with F-theory), so if you're in Heidelberg, he (or Timo Wiegand) might be people to ask. Of course, you propbably know that, so apologies... $\endgroup$ – Toffomat Aug 21 '17 at 16:46

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