Types of quantum field thories
There is an observation by Hořava–Witten 95, Hořava–Witten 96 which suggests that quantum 11-dimensional supergravity on an -orbifold (actually a higher orientifold) of the form induces on its boundary “M9-brane” (the -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 -action. Thereby the quantization condition
The supergravity C-field is supposed to vanish, and differentially vanish at the boundary in the HW model, meaning that also the local connection 3-form vanishes there. The argument is roughly as follows (similar for as in Falkowski, section 3.1).
Let be the canonical vector field along the circle factor. Then the component of which is annihilated by the contraction is necessarily even, so the component is also even. It follows that also is even.
Moreover, the kinetic term
is to be invariant. With the above this now implies that the components of annihiliated by is odd, because so is the mixed component of the metric tensor.
This finally implies that the restriction of to the orbifold fixed points has to be closed.
The original articles are
Reviews are in
Piyush Kumar, Hořava-Witten theory (2004) (pdf)