String Theory and M-Theory

10.4 Fluxes, torsion and heterotic strings 515
Once a solution for the hermitian Yang–Mills field has been found, the
fundamental form is constrained to satisfy the differential equation
i∂ ¯ ∂J = α′
8
[tr(R ∧ R) − tr(F ∧ F )] , (10.229)
which is a consequence of the anomaly cancellation condition.
To summarize, supersymmetry is unbroken if the external space-time is
Minkowski and the internal space satisfies the following conditions:
• It is complex and hermitian.
• There exists a holomorphic (3, 0)-form Ω that is related to the fundamental
form by the condition that the background is conformally balanced, that
is,
d(||Ω||J ∧ J) = 0. (10.230)
• The gauge field satisfies the hermitian Yang–Mills condition.
• The fundamental form satisfies the differential equation in Eq. (10.229).
These are the only conditions that have to be imposed. Once a solution
of the above constraints has been found, H and Φ are determined by the
data of the geometry according to
H = i(∂ − ¯ ∂)J and Φ = Φ0 − 1
log ||Ω||. (10.231)
2
There exist six-dimensional compact internal spaces that solve the above
constraints and lead to interesting phenomenological models in four dimensions.
However, they lie beyond the scope of this book. In the following we
describe a simpler example in which the internal space is four-dimensional.
Conformal K3
Four-dimensional internal spaces for heterotic-string backgrounds with torsion
can be constructed by considering an ansatz of the form of a direct
product in the string-frame, as before, with
gmn(y) = e 2D(y) g K3
mn(y), (10.232)
where g K3
mn(y) represents the (unknown) metric of K3, and gmn(y) is the
internal part of the string-frame metric. In this four-dimensional example,
the internal manifold is given by a conformal factor times a Calabi–Yau
manifold.