String Theory and M-Theory

546 Flux compactifications
complex structure defined by Eq. (10.29) is covariantly constant
∇pJm n = 0,
where ∇p is defined with respect to the metric gmn appearing in Eq. (10.5).
PROBLEM 10.3
Use the Fierz identity Eq. (10.318) to show that the almost complex structure
given in Eq. (10.29) satisfies J 2 = −1.
PROBLEM 10.4
Consider a flux compactification of M-theory on an eight manifold to threedimensional
Minkowski space-time. Suppose that two Majorana–Weyl spinors
of opposite chirality ξ+, ξ− on the eight-dimensional internal manifold
can be found
P±ξ = 1
2 (1 ± γ9 )ξ = ξ±,
so that the 8D spinor ξ = ξ+ + ξ− is nonchiral. Assuming that the internal
flux component is self-dual, show that, after an appropriate rescaling of the
spinor, the internal component of the gravitino supersymmetry transformation
takes the form
∇mξ+ − 1
4 ∆−3/4 Fmξ− = 0, ∇mξ− = 0.
PROBLEM 10.5
Consider M-theory compactified on an eight manifold with a nonchiral complex
spinor on the internal space. Recall that Eq. (10.17) showed that a
nonvanishing vector field can be constructed.
(i) Use the Fierz identity (10.318) to show that Eq. (10.17) implies that
the vector field relates the two (real) spinors of opposite chirality
η1 = v a γaη2.
(ii) Use part (i) and the result of Problem 10.4 to show that the primitivity
condition Eq. (10.36) is modified to
F ∧ J + ⋆ dv = 0,
where v has been rescaled by a constant.
PROBLEM 10.6
Show that the operations J3, J+, J− in Eq. (10.38) define an SU(2) algebra.