What are the masses M4 and M7 in terms of the mass parameter M ?

PROBLEM 8.3

Derive Eq. (8.69) and transform the bosonic part of the type IIA supergrav-

ity action in ten dimensions from the string frame to the Einstein frame.

16 Actually, in the quantum theory it has to be an integer multiple of a basic unit.

352 M-theory and string duality

PROBLEM 8.4

Derive the rede¬nitions of C1 , C3 , F2 and F4 that are required to display

a factor of e’2¦ 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 ¦ ’ ’¦ and

gµν ’ e’¦ 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 ¬elds in the

heterotic and type I theories map into one another to leading order in fermi

¬elds under an S-duality transformation, if » 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:

ρ2

ds = dρ + (dθ2 + dφ2 + dψ 2 ’ 2 cos θ dφ dψ)

2 2

4

with ψ ∼ ψ + 4π, and that this corresponds to a metric on ¬‚at four-

dimensional Euclidean space. Hint: let ψ = φ + 2y/R.

PROBLEM 8.10

Consider the ten-dimensional type IIA metric for a KK5-brane

5

2

dx2 + ds2 ,

ds = ’dt + i TN

i=1

where ds2 is given in Eqs (8.110) and (8.111).

TN

Homework Problems 353

(i) Use the rules presented in Section 6.4 to deduce the type IIB solution

that results from a T-duality transformation in the y direction.

(ii) What is the type IIB interpretation of the result?

(iii) Verify that the tension of the type IIB solution supports this inter-

pretation.

PROBLEM 8.11

In the presence of an M5-brane the 11-dimensional F4 satis¬es the Bianchi

identity

dF4 = δW ,

where δW is a delta function with support on the M5-brane world volume.

How must the equation of motion of F4 be modi¬ed in order to be compatible

with this Bianchi identity? What does this imply for the ¬eld content on

the M5-brane world volume? Hint: Consult Exercise 5.10.

PROBLEM 8.12

Verify that a type IIA D2-brane is an unwrapped M2-brane by showing that

TD2 = TM2 . Do the same for the NS5-brane and the M5-brane. Verify that

a wrapped M5-brane corresponds to a D4-brane.

PROBLEM 8.13

Verify Eqs (8.137) and (8.139).

PROBLEM 8.14

Verify that Eq. (8.139) implies that the Dirac quantization condition is sat-

is¬ed if the M2- and M5-brane each carry one unit of charge and saturate

the BPS bound.

PROBLEM 8.15

Derive Eqs (8.140), (8.141) and (8.142).

9

String geometry

Since critical superstring theories are ten-dimensional and M-theory is 11-

dimensional, something needs to be done to make contact with the four-

dimensional space-time geometry of everyday experience. Two main ap-

proaches are being pursued.1

Kaluza“Klein compacti¬cation

The approach with a much longer history is Kaluza“Klein compacti¬cation.

In this approach the extra dimensions form a compact manifold of size lc .

Such dimensions are essentially invisible for observations carried out at en-

ergy E 1/lc . Nonetheless, the details of their topology have a profound

in¬‚uence on the spectrum and symmetries that are present at low energies

in the e¬ective four-dimensional theory. This chapter explores promising

geometries for these extra dimensions. The main emphasis is on Calabi“Yau

manifolds, but there is also some discussion of other manifolds of special

holonomy. While compact Calabi“Yau manifolds are the most straight-

forward possibility, modern developments in nonperturbative string theory

have shown that noncompact Calabi“Yau manifolds are also important. An

example of a noncompact Calabi“Yau manifold, speci¬cally the conifold, is

discussed in this chapter as well as in Chapter 10.

Brane-world scenario

A second way to deal with the extra dimensions is the brane-world scenario.

In this approach the four dimensions of everyday experience are identi¬ed

with a defect embedded in a higher-dimensional space-time. This defect

1 Some mathematical background material is provided in an appendix at the end of this chapter.

Readers not familiar with the basics of topology and geometry may wish to study it ¬rst.

354

String geometry 355

is typically given by a collection of coincident or intersecting branes. The

basic fact (discussed in Chapter 6) that makes this approach promising is the

observation that Yang“Mills gauge ¬elds, like those of the standard model,

are associated with the zero modes of open strings, and therefore they reside

on the world volume of D-branes.

A variant of the Kaluza“Klein idea that is often used in brane-world sce-

narios is based on the observation that the extra dimensions could be much

larger than one might otherwise conclude if the geometry is warped in a

suitable fashion. In a warped compacti¬cation the overall scale of the four-

dimensional Minkowski space-time geometry depends on the coordinates of

the compact dimensions. This chapter concentrates on the more traditional

Kaluza“Klein approach, where the geometry is a product of an internal

manifold and an external manifold. Warped geometries and their use for

brane-world constructions are discussed in Chapter 10.

Motivation

The only manifolds describing extra dimensions that have been discussed

so far in this book are a circle and products of circles (or tori). Also, a

2 orbifold of a circle has appeared a couple of times. If any of the ¬ve