String Theory and M-Theory

508 Flux compactifications
where DaW = ∂aW + κ 2 4 ∂aK W . Thus
For small κ4,
V = e κ2
4K G a¯b DaW D¯b W − 3κ 2 4|W | 2
.
V = G a¯ b ∂aW ∂¯ b W + O(κ 2 4).
As expected, one finds the global supersymmetry formula plus corrections
proportional to Newton’s constant. ✷
10.4 Fluxes, torsion and heterotic strings
This section explores compactifications of the weakly coupled heterotic string
in the presence of a nonzero three-form field H. 20 A nonvanishing H flux has
two implications for the background geometry. First, the background geometry
becomes a warped product, like that discussed in the previous sections.
The second consequence of nonvanishing H is that its contributions to the
various equations can be given a geometric interpretation as torsion of the
internal manifold. If the gauge fields are not excited, heterotic supergravity
is a truncation of either type II supergravity theory. Therefore, some of the
analysis in this section applies to those cases and vice versa.
Warped geometry
As in the previous sections, when H flux is included, the space-time is no
longer a direct-product space of the form M10 = M4 ×M. (For simplicity, in
the following we assume that the external space-time is four-dimensional.)
Analysis of the heterotic supersymmetry transformation laws will show that
a warp factor e 2D(y) must be included in the metric in order to provide
a consistent solution. In the Einstein frame, let us write the background
metric for the warped compactification in the form
ds 2 = e 2D(y) (gµν(x)dx µ dx ν
+ gmn(y)dy m dy n
4D
6D
)
(10.194)
As before, x denotes the coordinates of the external space, y the internal
coordinates, the indices µ, ν label the coordinates of the external space and
m, n label the coordinates of the internal space.
The function D(y) depends only on the internal coordinates. It will be
shown that supersymmetry can be satisfied when there is nonzero H flux
provided that
20 The index on H3 is suppressed.
D(y) = Φ(y), (10.195)