Gravity and Strings

For the fermions:
16.4 The effective-field theory of the heterotic string 469
δɛψµ = ∂µ − 1
ωµ + 4
1
2 Hµσ 3 + k F (1) µσ 3 + k−1 F (2)
µ ɛ
δɛρ = 1
2
δɛλ =
1 (−1)n
k 2 G
n! (n) ƔµP[ n 2 ]+1ɛ.
1 (1) 3 −1 (2) ∂ ln k + k F σ − k F 8
ɛ
+ 1 8 1 (−1)n
eφ k 2 G
16 i=1 n! (n) P[ n 2 ]+1ɛ,
∂φ − 1
12 Hσ 3 − 1
(1) 3 −1 (2) k F σ + k F 8
ɛ
+ 1 8
eφ
16 i=1
+ 1
16
eφ 8
i=1
n (9 − 2n)
(−1)
n!
k (−1)n
2 G (n) P[ n 2 ]+1ɛ.
16.4 The effective-field theory of the heterotic string
(16.90)
The full action and supersymmetry transformation rules of the N = 2A, d = 10 supergravity
theory are invariant under the two Z2 transformations:
Ĉ (2n+1) →−Ĉ (2n+1) , ˆψ ˆµ →±ˆƔ11 ˆψ ˆµ, ˆλ →∓ˆƔ11 ˆλ, ˆɛ →±ˆƔ11 ˆɛ. (16.91)
These transformations correspond to the 11-dimensional transformation of the compact
coordinate15 z →−z combined with the transformation f →±f for all the fermions of
the theory, which is always a symmetry. Eliminating all the fields which are odd under
these transformations is always a consistent truncation of N = 2A, d = 10 supergravity
that is equivalent, according to the discussion in Section 11.6, to the compactification of
11-dimensional supergravity on the orbifold S1 /Z2. Inthe two possible truncations all the
RR fields and half of the fermions (a chiral half) are eliminated. The result is a chiral theory
that is invariant under supersymmetry transformations generated by a single Majorana–
Weyl fermion, i.e. N = 1, d = 10 supergravity. The action for the bosonic sector of this
theory is that of the common sector Eq. (15.1). As for the supersymmetry transformation
rules, defining for all fermions fˆ
f ˆ = f ˆ(+)
+ f ˆ(−)
,
15 One should remember that the 11-dimensional Ĉ is a pseudotensor.
ˆµˆν ˆρ
ˆƔ11 ˆ
f (±) ≡± ˆ
f (±) , (16.92)