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Through timelike dualities, one can generate exotic versions of $M$-theorywith different spacetime signatures. These are the $M^*$-theory with signature$(9,2,-)$, the $M'$-theory, with signature $(6,5,+)$ and the theories withreversed 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 absoluteparallelism are, besides Lie groups, the seven-sphere $S^7 \equiv SO(8)/SO(7)$and its pseudo-riemannian version $S^{3,4} \equiv SO(4,4)/SO(3,4)$. [There isalso the complexification $SO(8,\mathbb{C})/SO(7, \mathbb{C})$, but it is ofdimension too high for our considerations.] The seven-sphere $S^7\equivS^{7,0}$ has been found to play an important role in $11$-dimensionalsupergravity, both through the Freund-Rubin solution and the Englert solutionthat uses its remarkable parallelizability to turn on non trivial internalfluxes. The spacetime manifold is in both cases $AdS_4 \times S^7$. We showthat $S^{3,4}$ enjoys a similar role in $M'$-theory and construct the exoticform $AdS_4 \times S^{3,4}$ of the Englert solution, with non zero internalfluxes turned on. There is no analogous solution in $M^*$-theory.

Timelike duality, $M'$-theory and an exotic form of the Englert solution