Mathematical Stmctures 149(which is the torsion in the heterotic theory) will play a central role in non-Kahler
stuctures.
More importantly, the study of complex non-Kahler manifolds is another step
in understanding the whole space of string solutions or vacua. The space of string
vacua contains both geometrical and non-geometrical regions. But even within
the geometrical region, the compactification manifold need not be Kahler nor even
complex (for type IIA theory [9]) when a’ corrections and branes are allowed. This
seems to give many possibilities for the geometry of the internal six-manifold for
different types of string theories. However, since the six different string theories
are related to each other through various dualities, the geometries and structures
of six-dimensional compact manifolds associated with string vacua are most likely
also subtlely related. This gives hope that the space of string vacua can indeed be
understood well-enough such that we can confirm or rule out that there exists at
least one string vacuum that can reproduce the four-dimensional standard model of
our world. Given the recent successes of compactification with fluxes - from moduli
fixing [lo] to addressing the cosmological constant issue [ll] - we can expect that
the physical real world vacuum will involve fluxes and understanding the structures
of non-Kahler manifolds may prove indispensable.
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