Gravity and Strings

16.5 Toroidal compactification of the heterotic string 477
These supersymmetry transformation rules are clearly covariant under O(n, n) T-duality
transformations. This means that any d-dimensional solution will have the same number of
unbroken supersymmetries after an O(n, n) rotation. The corresponding ten-dimensional
solutions may but need not have the same amount of supersymmetry. This is due to the
fact that unbroken supersymmetry can be broken in dimensional reduction. We are going to
discuss this subtle point in Section 16.6, but obviously it applies to many other situations:
for instance the relation between the unbroken supersymmetries of N = 1, d = 11 and N =
2A, d = 10 supergravity solutions.
16.5.3 The truncation to pure supergravity
The fields of the reduced theory correspond to pure supergravity (16 supercharges in d
dimensions) coupled to n vector supermultiplets. The fields in the supergravity multiplet
are the gravitinos ˆψ (+)
a , the dilatino ˆλ (−) − ˆƔ i ˆψ (+)
i , the graviton ea µ, the dilaton φ, the KR
2-form Bµν, and n of the 2n KK and winding vectors A . The n vector supermultiplets are
made out of the n2 scalars contained in V A , n of the 2n KK and winding vectors A , and
the gauginos ˆψ (+) A2. Thus, we know to which supermultiplet each field belongs, except for
the vector fields. These, however, can be identified by studying the truncation of the vector
multiplets, which consists in
E = In×n, B = 0, ˆψ (+) A2 = 0, (16.136)
plus the vanishing of the matter vector fields. Since the truncation has to be consistent at
the level of the equations of motion, if we substitute the above values of the fields into the
equations of motion of the theory, we will be forced to set to zero n combinations of the 2n
vector fields A , which are then identified with the matter vector fields. The n orthogonal
combinations that remain are the supergravity vector fields.
Substituting M = I2n×2n into Eq. (16.128) tells us only that F 1F 2 = 0, though, and
we also have to impose consistency of the truncation of the supersymmetry transformation
rules Eqs. (16.135). On substituting V = 0andδˆɛ ˆψ (+) A2 = 0, we find
F 2 = 0 ⇒− 1
√2 (F 1 − F 2 ) = 0, (16.137)
which implies that the combinations F 2 are the matter vector fields and the F 1 are the
supergravity ones.
The action of the truncated pure supergravity d-dimensional theory is
S =
g2 h
16πG (d)
N
dd x √ |g| e−2φ
R − 4(∂φ) 2 + 1
2 · 3! H 2 − 1
4
F 1F 2
, (16.138)