p’1

(’1) 2p!q!

’ 3!(p’3)!(q’3)! δ[m1 m2 m3 [n1 n2 n3 γm4 ...mp ] n4 ...nq ]

+...

The Fierz transformation identity for commuting spinors is

d

1 1 mp ...m1 ¯

¯

χψ = γ ψγm1 ...mp χ. (10.318)

2[d/2] p!

p=0

In the case of anticommuting spinors there is an additional minus sign.

HOMEWORK PROBLEMS

PROBLEM 10.1

Show that covariant derivatives with respect to conformally transformed

metrics gM N = „¦2 gM N are related by

ˆ

1

ˆ M· = · + „¦’1 N

N „¦“M ·.

M

2

Use this result to derive Eq. (10.18).

PROBLEM 10.2

Re-express the supersymmetry transformation Eq. (10.26) in terms of the

rescaled spinor ξ = ∆1/4 ·. Use this equation to show that the almost

546 Flux compacti¬cations

complex structure de¬ned by Eq. (10.29) is covariantly constant

n

p Jm = 0,

where is de¬ned with respect to the metric gmn appearing in Eq. (10.5).

p

PROBLEM 10.3

Use the Fierz identity Eq. (10.318) to show that the almost complex struc-

ture given in Eq. (10.29) satis¬es J 2 = ’1.

PROBLEM 10.4

Consider a ¬‚ux compacti¬cation of M-theory on an eight manifold to three-

dimensional Minkowski space-time. Suppose that two Majorana“Weyl spin-

ors of opposite chirality ξ+ , ξ’ on the eight-dimensional internal manifold

can be found

1

P± ξ = (1 ± γ 9 )ξ = ξ± ,

2

so that the 8D spinor ξ = ξ+ + ξ’ is nonchiral. Assuming that the internal

¬‚ux component is self-dual, show that, after an appropriate rescaling of the

spinor, the internal component of the gravitino supersymmetry transforma-

tion takes the form

1 ’3/4

m ξ+ ’ ∆ Fm ξ’ = 0, m ξ’ = 0.

4

PROBLEM 10.5

Consider M-theory compacti¬ed on an eight manifold with a nonchiral com-

plex spinor on the internal space. Recall that Eq. (10.17) showed that a

nonvanishing vector ¬eld can be constructed.

(i) Use the Fierz identity (10.318) to show that Eq. (10.17) implies that

the vector ¬eld relates the two (real) spinors of opposite chirality

·1 = v a γa ·2 .

(ii) Use part (i) and the result of Problem 10.4 to show that the primi-

tivity condition Eq. (10.36) is modi¬ed 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) de¬ne an SU (2) algebra.

Homework Problems 547

PROBLEM 10.7

Verify Eq. (10.226), which shows that the ¬‚ux 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 ge-

ometry, type IIB theories compacti¬ed 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 compacti¬ed on a Calabi“Yau three-fold in the presence of ¬‚uxes is

given by

¯

V = eK Gab Da W D¯W ’ 3|W |2 ,

b

where

„¦ § G3 .

W=

M

Here a, b label all the holomorphic moduli. You can assume that the K¨hler

a

potential is given by Eq. (10.104).

PROBLEM 10.10

Show that, in a Calabi“Yau four-fold compacti¬cation of M-theory, the sta-

tionary points of

| γ „¦|2

2

|Z(γ)| = ¯

„¦§„¦

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 Poincar´ dual cycle to the four-form F . A related result is derived in

e

Chapter 11 in the context of the attractor mechanism for black holes.

PROBLEM 10.11

Show that the Christo¬el 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

548 Flux compacti¬cations

Eq. (10.86). In order to do this, you have to take into account the ¬rst ±

correction to the D7-brane action, whose form is given in Eq. (10.91) after

determining the coe¬cient.

PROBLEM 10.13

Show that the K¨hler form J of the singular conifold described in Sec-

a

tion 10.2 can be written in terms of a basis of one-forms according to

2 12

e § e1 + e3 § e4

dr § g5 +

J=

3 3

Deduce that G3 , given by Eqs (10.133) and (10.135), is primitive.

PROBLEM 10.14

For the heterotic string with torsion there is an identity of the form

= ’e’a¦ d(ea¦ J).

6H

Derive the value of the parameter a for which this is true.