A paper by Lalak et al, entitled "Soliton Solutions of M-theory on an orbifold", considers the brane solutions of 11 dimensional supergravity on a space of the form $R^{10} \times S^1/\mathbb{Z}_2$.

If $x^{11}$ is the orbifolded coordinate, such that $x^{11} \in [-\pi \rho, \pi \rho]$ and one identifies $x^{11} \rightarrow -x^{11}$, then essentially there are two 10 dimensional hyperplanes at $x^{11} = 0$ and $x^{11} = \pi\rho$.

Suppose $A, B, \ldots$ denote 10 dimensional indices, and the fields of 11D supergravity are: the metric $g$, the 3-form tensor field $C_{(3)}$, and the gravitino.

Then the paper states that the following conditions must be imposed on the metric and the gauge fields and the field strength, so that the solutions to the equations of motion are the ones that respect the $\mathbb{Z}_2$ orbifold symmetry.

$$g_{AB}(x^{11}) = g_{AB}(-x^{11})$$ $$G_{ABCD}(x^{11}) = -G_{ABCD}(-x^{11})$$ $$g_{A11}(x^{11}) = -g_{A11}(-x^{11})$$ $$G_{11BCD}(x^{11}) = G_{11BCD}(-x^{11})$$ $$g_{11,11}(x^{11}) = g_{11,11}(-x^{11})$$ $$C_{ABC}(x^{11}) = -C_{ABC}(-x^{11})$$ $$C_{11BC}(x^{11}) = C_{11BC}(-x^{11})$$

The constraints on the form fields should follow from the invariance of the Chern-Simons action under orbifolding, but what is the exact argument?

This post imported from StackExchange Physics at 2015-07-07 10:59 (UTC), posted by SE-user leastaction