Timelike duality, $M^{\mathrm{\prime }}$-theory and an exotic form of the Englert solution

Through timelike dualities, one can generate exotic versions of $M$-theory with different spacetime signatures. These are the $M^{\ast }$-theory with signature $(9,2,-)$, the $M^{\mathrm{\prime }}$-theory, with signature $(6,5,+)$ and the theories with reversed signatures $(1,10,-)$, $(2,9,+)$ and $(5,6,-)$. In $(s,t,\pm )$, $s$ is the number of space directions, $t$ the number of time directions, and $\pm$ refers to the sign of the kinetic term of the $3$ form. The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere $S^7\equiv SO(8)\mathrm{/}SO(7)$ and its pseudo-riemannian version $S^{3,4}\equiv SO(4,4)\mathrm{/}SO(3,4)$. [There is also the complexification $SO(8,\mathbb{C})\mathrm{/}SO(7,\mathbb{C})$, but it is of dimension too high for our considerations.] The seven-sphere $S^7\equiv S^{7,0}$hasbeenfoundtoplayanimportantrolein$11$-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases $Ad{S}_4\times S^7$. We show that $S^{3,4}$ enjoys a similar role in $M^{\mathrm{\prime }}$-theory and construct the exotic form $Ad{S}_4\times S^{3,4}$ of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in $M^{\ast }$-theory.