String Theory and M-Theory

SOLUTION
10.4 Fluxes, torsion and heterotic strings 517
In order to prove that the manifold is complex one computes the Nijenhuis
tensor, which was defined in the appendix of Chapter 9 to be
Nmn p = Jm q J [n p ,q] − Jn q J [m p ,q].
Eq. (10.217) implies that the Nijenhuis tensor takes the form
Nmnp = 1
q
Hmnp − 3J [m Jn
2
s
Hp]qs Identities for Dirac matrices, which are listed in the appendix of this chapter,
imply
J p
[m Jn]
q 1 = 4gprgqs (J ∧ J)mrns + 1
2JmnJ pq
= 1
2η† γpq mnη − 1
2η† γpqη η † γmnη ,
where the last line has used the six-dimensional identity
1
(J ∧ J) = ∗J.
2
As a result, one obtains
Nmnp = − 1
12η†
+ H, γmnp + 3iγ [mJnp] η+
= − 1
12η†
+ /∂Φ, γmnp + 3iγ [mJnp] η+
= 0.
This proves that the manifold is complex. ✷
EXERCISE 10.11
Prove that Ω in Eq. (10.222) is holomorphic.
SOLUTION
A holomorphic three-form is a ¯ ∂ closed form of type (3, 0). In order to prove
that Ω is holomorphic, we compute ¯ ∂Ω. We start by computing its covariant
derivative.
The covariant derivative (defined with respect to the Christoffel connection)
acting on the tensor Ω is
∇¯ k Ωabc = ∂¯ k Ωabc − 3Γ p ¯ k[a Ω bc]p = ∂¯ k Ωabc − Γ p ¯ kp Ωabc.
Using the definition of the Christoffel connection and expanding Eq. (10.220)
in components implies
Γ p ¯ kp = g p¯q ∂ [ ¯ k g ¯q]p = 1
2 H¯ kp¯q gp¯q = ∂¯ k Φ.