Mathematical Structures^1474.5.2 Shing- Tung Yau: Mathematical Structures: Geometry of
Six-Dimensional String
There has been a great deal of anxiety to find a suitable string vacuum solution
or to perform statistics over the space of all such vacua. However, despite great
successes in the twenty years since the first string revolution, our understanding of
string vacua is far from complete. For starters, we have not achieved a satisfactory
theory of computing supersymmetric cycles nor a good understanding of Hermitian-
Yang-Mills fields and their instantons. Such issues pertain to deep problems in
mathematics and ideas inspired from physical considerations have been essential
for progress.
In the compactificaton of Candelas-Horowitz-Strominger-Witten [3], preserving
supersymmetry with zero H-flux requires the compact six-manifold to be Kahler
Calabi-Yau. While this class of manifold is quite large, it is believed to have a finite
number of components with finite dimensions for its moduli space.
For the class of three-dimensional Kahler Calabi-Yau manifolds, there is a con-
struction due to works of Clemens [4] and Friedman [6] where one takes a finite
number of rational curves with negative normal bundle and pinch them to coni-
fold points. Under suitable conditions for the homology class, one can deform the
resulting (singular) manifold to a smooth manifold. The resulting manifold is in
general non-Kahler. By repeating such procedures several times, one can obtain a
smooth complex manifold with vanishing second Betti number (and hence clearly
non-Kahler). If the homology of the original Calabi-Yau manifold has no torsion,
a theorem of Wall [16] shows that the resulting manifold must be diffeomorphic to
a connected sums of S3 x S3. This type of manifold can be considered as a nat-
ural generalization of Riemann surfaces which are connected sums of handle bodies.
These three-dimensional complex manifold also have a holomorphic three-form that
is naturally inherited from the original Calabi-Yau.
There is a proposal of Reid [14] that the moduli space of all Calabi-Yau struc-
tures can be connected through such complex structures over handle bodies. Such
a proposal may indeed be true. However, an immediate problem is that we are
then required to analyze non-Kahler complex manifolds but we have virtually no
theory for them. Non-Kahler manifolds have appeared naturally in string compact-
ifications with fluxes. So perhaps a useful way to think about the construction of
Clemens and Friedman is that the collapsing of the rational curves together with
the deformation of the complex structure correspond to turning on a flux. Hence,
just from mathematical considerations of the Calabi-Yau moduli space, we are led
to study structures which contain fluxes and preserve supersymmetries.
A natural supersymmetric geometry with flux to consider is the one in heterotic
string theory. The geometry is constrained by a system of differential equations
worked out by Strominger [15] and takes the following form
411 0 IIW w^2 1 = 0 ,
F2>' = = 0 , F p, w2 = 0 ,