String Theory and M-Theory

9.4 Calabi–Yau compactifications of the heterotic string 375
where the scalar curvature R = g µρ g νλ Rµνρλ is a constant. It is proportional
to the four-dimensional cosmological constant. Maximal symmetry restricts
the space-time M4 to be either Minkowski (R = 0), AdS (R < 0) or dS
(R > 0). The assumption of maximal symmetry along M4 also requires
the following components of the NS–NS three-form field strength H and the
Yang–Mills field strength to vanish
Hµνρ = Hµνp = Hµnp = 0 and Fµν = Fµn = 0. (9.43)
In this chapter it is furthermore assumed that the internal three-form field
strength Hmnp vanishes and the dilaton Φ is constant. These assumptions,
made for simplicity, give rise to the backgrounds described in this chapter.
Backgrounds with nonzero internal H-field and a nonconstant dilaton are
discussed in Chapter 10.
Conditions for unbroken supersymmetry
The constraints that N = 1 supersymmetry imposes on the vacuum arise
in the following way. Each of the supersymmetry charges Qα generates an
infinitesimal transformation of all the fields with an associated infinitesimal
parameter εα. Unbroken supersymmetries leave a particular background
invariant. This is the classical version of the statement that the vacuum
state is annihilated by the charges. The invariance of the bosonic fields
is trivial, because each term in the supersymmetry variation of a bosonic
field contains at least one fermionic field, but fermionic fields vanish in a
classical background. Therefore, the only nontrivial conditions come from
the fermionic variations
δε(fermionic fields) = 0. (9.44)
In fact, for exactly this reason, only the bosonic parts of fermionic supersymmetry
transformations were presented in Chapter 8. If the expectation
values for the fermions still vanish after performing a supersymmetry variation,
then one obtains a solution of the bosonic equations of motion that
preserves supersymmetry for the type of backgrounds considered here. In
fact, as is shown in Exercise 9.4, a solution to the supersymmetry constraints
is always a solution to the equations of motion, while the converse is not
necessarily true. Here we are applying this result for theories with local
supersymmetry. This can be done if we impose the Bianchi identity satisfied
by the three-form H as an additional constraint. In order to obtain
unbroken N = 1 supersymmetry, Eq. (9.44) needs to hold for four linearly