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There is an observation by Hořava–Witten 95, Hořava–Witten 96 which suggests that quantum 11-dimensional supergravity on an $\mathbb{Z}_2$-orbifold (actually a higher orientifold) of the form $X_{10} \times /(S^1//\mathbb{Z}_2))$ induces on its boundary “M9-brane” (the $\mathbb{Z}_2$-fixed point manifold) heterotic string theory. Therefore one also speaks of “heterotic M-theory” (Ovrut 02).
The orbifold equivariance condition of the supergravity C-field is that discussed at orientifold (there for the B-field). Therefore it has to vanish at the two fixed fixed points of the $\mathbb{Z}_2$-action. Thereby the quantization condition
on the supergravity C-field becomes the condition for the Green-Schwarz mechanism of the heterotic string theory on the “boundary” (the orbifold fixed points).
The supergravity C-field $\hat G_4$ is supposed to vanish, and differentially vanish at the boundary in the HW model, meaning that also the local connection 3-form $C_3$ vanishes there. The argument is roughly as follows (similar for as in Falkowski, section 3.1).
in the Lagrangian of 11-dimensional supergravity is supposed to be well-defined on fields on the orbifold and hence is to be $\mathbb{Z}_2$-invariant.
Let $\iota_{11}$ be the canonical vector field along the circle factor. Then the component of $G \wedge G$ which is annihilated by the contraction $\iota_{11}$ is necessarily even, so the component $d x^{11}\wedge \iota_11 C_3$ is also even. It follows that also $d x^{11}\wedge \iota_11 G_4$ is even.
Moreover, the kinetic term
is to be invariant. With the above this now implies that the components of $G$ annihiliated by $\iota_{11}$ is odd, because so is the mixed component of the metric tensor.
This finally implies that the restriction of $C_3$ to the orbifold fixed points has to be closed.
The original articles are
Petr Hořava, Edward Witten, Eleven dimensional supergravity on a manifold with boundary, Nucl. Phys. B475 (1996) 94 (arXiv:hep-th/9603142)
Reviews are in
Piyush Kumar, Hořava-Witten theory (2004) (pdf)
Paul Townsend, Four Lectures on M-Theory (arXiv:hep-th/9612121).
Burt Ovrut, Lectures on Heterotic M-Theory (arXiv:hep-th/0201032)
Adam Falkowski, section 3 of Five dimensional locally supersymmetric theories with branes, Master Thesis, Warsaw (pdf)
Explicit discussion of M2-branes ending on the HW fixed points and becoming heterotic strings there is discussed, via the BLG model, in
After KK-reduction to 5d supergravity there is a corresponding 5d mechanism, see the references there.