String Theory and M-Theory

Homework Problems 547
PROBLEM 10.7
Verify Eq. (10.226), which shows that the flux backgrounds for the weakly
coupled heterotic string in Section 10.4 are conformally balanced.
PROBLEM 10.8
Show that, in the absence of sources or singularities in the background geometry,
type IIB theories compactified to four dimensions do not admit dS
space-times as solutions to the equations of motion. In other words, repeat
the computation that led to Eq. (10.86) by allowing a cosmological constant
Λ in external space-time.
PROBLEM 10.9
Assuming a constant dilaton, show that the scalar potential of type IIB
theory compactified on a Calabi–Yau three-fold in the presence of fluxes is
given by
where
K
V = e G a¯b DaW D¯b W − 3|W | 2
,
W =
M
Ω ∧ G3.
Here a, b label all the holomorphic moduli. You can assume that the Kähler
potential is given by Eq. (10.104).
PROBLEM 10.10
Show that, in a Calabi–Yau four-fold compactification of M-theory, the stationary
points of
|Z(γ)| 2
| γ
= Ω|2
Ω ∧ Ω¯
are given by the points in moduli space where |Z(γ)| 2 = 0, or if |Z(γ)| 2 =
0, then F has to satisfy F 1,3 = F 3,1 = 0. In the above expression γ is
the Poincaré dual cycle to the four-form F . A related result is derived in
Chapter 11 in the context of the attractor mechanism for black holes.
PROBLEM 10.11
Show that the Christoffel connection does not transform as a tensor under
coordinate transformations, but that torsion transforms as a tensor.
PROBLEM 10.12
Show that D7-branes give a negative contribution to the right-hand side of