String Theory and M-Theory

352 M-theory and string duality
PROBLEM 8.4
Derive the redefinitions of C1, C3, F2 and F4 that are required to display
a factor of e −2Φ in the terms SR and SCS of the type IIA action given in
Eqs (8.41) and (8.42).
PROBLEM 8.5
Show that SCS in Eq.(8.42) is invariant under a U(1) gauge transformation
even though it contains F4 rather than ˜ F4.
PROBLEM 8.6
Consider the type IIB bosonic supergravity action in ten dimensions given
in Eq. (8.53). Setting C0 = 0, perform the transformations Φ → −Φ and
gµν → e −Φ gµν. What theory do you obtain, and what does the result imply?
How should the transformations be generalized when C0 = 0?
PROBLEM 8.7
Verify that the actions in Eqs (8.73) and (8.81) map into one another under
the transformations (8.88) and (8.89).
PROBLEM 8.8
Verify that the supersymmetry transformations of the fermi fields in the
heterotic and type I theories map into one another to leading order in fermi
fields under an S-duality transformation, if λ and χ are suitably rescaled.
PROBLEM 8.9
Consider the Euclidean Taub–NUT metric (8.110). Show that there is no
singularity at r = 0 by showing that the metric takes the following form
near the origin:
ds 2 = dρ 2 + ρ2
4 (dθ2 + dφ 2 + dψ 2 − 2 cos θ dφ dψ)
with ψ ∼ ψ + 4π, and that this corresponds to a metric on flat fourdimensional
Euclidean space. Hint: let ψ = φ + 2y/R.
PROBLEM 8.10
Consider the ten-dimensional type IIA metric for a KK5-brane
where ds 2 TN
ds = −dt 2 +
5
i=1
dx 2 i + ds 2 TN,
is given in Eqs (8.110) and (8.111).