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Roger Penrose in "The Road to Reality" makes a devastating refutation of
the extra space dimensions of string theory on purely mathematical
grounds. The extra space dimensions come from there being a faster than
light tachyon in the string theory in 3 space dimensions. However, that
simply means that some continuous symmetry of the action of the string
is spontaneously broken inducing a vacuum phase transition of the theory
with macro-quantum ODLRO - emergence of a Higgs-type inflation field
with one or more Goldstone phases like in a superconductor. Therefore, a
string theory in ordinary 3 + 1 space-time may be possible after all.
Indeed, curved space-time's renormalizable spin 1 tetrad quantum fields
and spin-connections with dark energy and dark matter seem to naturally
emerge from the Goldstone phases induced by the tachyon instability. The
spin 2 metric field is an emergent entangled composite of pairs of spin
1 tetrad fields and torsion fields.
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