148 The Quantum Structure of Space and TimedH = aaaw = a'(trRA R- trF A F) ,
where w is the hermitian metric, R the holomorphic three-form, and F
the Hermitian-Yang-Mills field strength. In above, the H-flux is given by
It would be nice to understand geometrically how the flux can be turned on from
a thorough analysis of the Strominger system for the non-Kahler Calabi-Yau handle
bodies. As a first step, Li-Yau [12] have shown in a rather general setting, that one
can always obtain a solution to the above equations by perturbing the Calabi-
Yau vacuum with the gauge bundle being a sum of the tangent bundle together
with copies of the trivial bundle. The deformation to non-zero H-flux will mix
together the tangent and trivial bundle parts of the gauge bundle. This allows Li-
Yau to construct non-zero H-flux solutions with SU(4) and SU(5) gauge group. In
the analysis of the deformations of such gauge bundles, the deformation space of
the Kahler and complex structure of the Calabi-Yau naturally arised. Therefore,
studying such deformation to non-zero H-flux systems can give insights into the
moduli space of Calabi-Yau.
The first equation of the Strominger system calls for the existence of a balanced
metric on such manifolds. These are n-dimensional complex manifolds which admit
a hermitian metric w that satisfies d(wn-') = 0 [13]. Balanced metrics satisfy many
interesting properties such as being invariant under birational transformations as
was observed by Alessandrini and Bassanelli [ 11. Using parallel spinors, it is possible
to decompose the space of differential forms similar to that of Hodge decomposition.
This has been carried out by my student C. C. Wu in her thesis.
Presently, we do not know how large is the class of balanced manifolds. Michel-
sohn has shown that for the twistor space of anti-self-dual four manifolds, the natural
complex structure is balanced [13]. It may be useful to identify such manifolds whose
anti-canonical line bundle admits a holomorphic three-form. Another well-known
class of non-Kahler manifolds that is balanced consists of K3 surfaces fibered with
a twisted torus bundle. In this special case, there is a metric ansatz [5, 81 which
enabled Fu-Yau [7] to demonstrate the existence of a solution to the Strominger
system that is not connected to a Calabi-Yau manifold. The existence of such a
solution is consistent with duality chasing arguments from M-theory that were first
discussed in detail by Becker-Dasgupta [2].
As mentioned, the theory of complex non-Kahler manifolds has not been devel-
oped much. Similar to Calabi-Yau compactification, it will be important to rephrase
the four-dimensional physical quantities like the types and number of massless fields
or the Yukawa coupling in terms of the properties of the non-Kahler manifold. For
example, can the number massless modes or geometric moduli be expressed purely
in terms of certain geometrical quantities perhaps analogous to the Hodge numbers
for the Kahler case. Here, trying to answer such physical questions will compel us
to seek a deeper understanding of the differential structures of non-Kahler mani-
fold than that known currently. It is likely that fluxes and in particular the H-flux
H = a- y(a - 8)~.