Horava-Witten Theory

Just as the M-theory compactification on ${\bf S}^1_R$ leads to the Type IIA theory, Horava and Witten realized that orbifold compactifications leads to the $E_8 \times E_8$ heterotic theory in ten dimensions HE (see for instance [23]). More precisely

$${\rm M}/({\bf S}^1/{\bf Z}_2) \Longleftrightarrow HE$$ (56)

where ${\bf S}^1/ {\bf Z}_2$ is homeomorphic to the finite interval $I$ and the $M$-theory is thus defined on $Y = X\times I$. From the ten-dimensional point of view, this configuration is seen as two parallel planes placed at the two boundaries $\partial I$ of $I$. Dimensional reduction and anomalies cancellation conditions imply that the gauge degrees of freedom should be trapped on the ten-dimensional planes $X$ with the gauge group being $E_8$ in each plane. While that the gravity is propagating in the bulk and thus both copies of $X$'s are only connected gravitationally. truecm