Homework Problems 295

where the coordinates x, y, z each have period 2π. Suppose there is also a

nonvanishing three-form Hxyz = N , where N is an integer. For example,

Bxy = N z.

(i) Using the T-duality rules for background ¬elds derived in Chapter 6,

carry out a T-duality transformation in the x direction followed by

another one in the y direction. What is the form of the resulting

metric and B ¬elds?

(ii) One can regard the T 3 as a T 2 , parametrized by x and y, ¬bered over

the z-circle. Going once around the z-circle is trivial in the original

background. What happens when we go once around the z-circle

after the two T-dualities are performed?

(iii) The background after the T-dualities has been called nongeometrical.

Explain why. Hint: use the results of the preceding problem.

PROBLEM 7.14

Consider the compacti¬cation of each of the two supersymmetric heterotic

string theories on a circle of radius R. As discussed in Section 7.4, the

moduli space is 17-dimensional and at generic points the left-moving gauge

symmetry is U (1)17 . However, at special points there are enhanced sym-

metries. Assume that the gauge ¬elds in the compact dimensions, that is,

the Wilson lines, are chosen in each case to give SO(16) — SO(16) — U (1)

left-moving gauge symmetry. Show that the two resulting nine-dimensional

theories are related by a T-duality transformation that inverts the radius of

the circle. This is very similar to the T-duality relating the type IIA and

IIB superstring theories compacti¬ed on a circle.

PROBLEM 7.15

(i) Compactifying the E8 — E8 heterotic string on a six-torus to four

dimensions leads to a theory with N = 4 supersymmetry in four

dimensions. Verify this statement and assemble the resulting massless

spectrum into four-dimensional supermultiplets.

(ii) Repeat the analysis for the type IIA or type IIB superstring. What is

the amount of supersymmetry in four dimensions in this case? What

is the massless supermultiplet structure in this case?

8

M-theory and string duality

During the “Second Superstring Revolution,” which took place in the mid-

1990s, it became evident that the ¬ve di¬erent ten-dimensional superstring

theories are related through an intricate web of dualities. In addition to

the T-dualities that were discussed in Chapter 6, there are also S-dualities

that relate various string theories at strong coupling to a corresponding dual

description at weak coupling. Moreover, two of the superstring theories (the

type IIA superstring and the E8 — E8 heterotic string) exhibit an eleventh

dimension at strong coupling and thus approach a common 11-dimensional

limit, a theory called M-theory. In the decompacti¬cation limit, this 11-

dimensional theory does not contain any strings, so it is not a string theory.

Low-energy e¬ective actions

This chapter presents several aspects of M-theory, including its low-energy

limit, which is 11-dimensional supergravity, as well as various nonpertur-

bative string dualities. Some of these dualities can be illustrated using

low-energy e¬ective actions. These are supergravity theories that describe

interactions of the massless ¬elds in the string-theory spectrum. It is not

obvious, a priori, that this should be a useful approach for analyzing nonper-

turbative features of string theory, since extrapolations from weak coupling

to strong coupling are ordinarily beyond control. However, if one restricts

such extrapolations to quantities that are protected by supersymmetry, one

can learn a surprising amount in this way.

BPS branes

A second method of testing proposed duality relations is to exploit the

various supersymmetric or Bogomolny“Prasad“Sommer¬eld (BPS) p-branes

296

M-theory and string duality 297

that these theories possess and the matching of the corresponding spectra of

states. As we shall illustrate below, saturation of a BPS bound can lead to

shortened supersymmetry multiplets, and then reliable extrapolations from

weak coupling to strong coupling become possible. This makes it possible to

carry out detailed matching of p-branes and their tensions in dual theories.

The concept of a BPS bound and its saturation can be illustrated by mas-

sive particles in four dimensions. The N -extended supersymmetry algebra,

restricted to the space of particles of mass M > 0 at rest in D = 4, takes

the form

{QI , Q†J } = 2M δ IJ δ±β + 2iZ IJ “0 , (8.1)

± ±β

β

where Z IJ is the central-charge matrix. I, J = 1, . . . , N labels the super-

symmetries and ±, β = 1, 2,3,4 labels the four components of each Majorana

spinor supercharge. The central charges are conserved quantities that com-

mute with all the other generators of the algebra. They can appear only

in theories with extended supersymmetry, that is, theories that have more

supersymmetry than the minimal N = 1 case, because the central-charge

matrix is antisymmetric Z IJ = ’Z JI . The central charges are electric and

magnetic charges that couple to the gauge ¬elds belonging to the supergrav-

ity multiplet.

By a transformation of the form Z ’ U T ZU , where U is a unitary matrix,

the antisymmetric matrix Z IJ can be brought to the canonical form

«

0 Z1 0 0

¬ ’Z1 0 0 ... ·

0

¬ ·

¬0 ·

0 0 Z2

IJ

Z =¬ (8.2)

·

¬0 ·

0 ’Z2 0

. ..

. .

.

with |Z1 | ≥ |Z2 | ≥ . . . ≥ 0. The structure of Eq. (8.1) implies that the

2N — 2N matrix

MZ

(8.3)

Z† M

should be positive semide¬nite. This in turn implies that the eigenvalues

M ±|Zi | have to be nonnegative. Therefore, the mass is bounded from below

by the central charges, which gives the BPS bound

M ≥ |Z1 |. (8.4)

States that have M = |Z1 | are said to saturate the BPS bound. They be-

long to a short supermultiplet or BPS representation. States with M > |Z1 |

298 M-theory and string duality

belong to a long supermultiplet. The zeroes that appear in the supersym-

metry algebra when M = |Z1 | are responsible for the multiplet shortening.

A further re¬nement in the description of BPS states keeps track of the

number of central charges that equal the mass. Thus, for example, in the

N = 4 case, states with M = |Z1 | = |Z2 | are called half-BPS and ones with

M = |Z1 | > |Z2 | are called quarter-BPS. These fractions refer to the number

of supersymmetries that are unbroken when these particles are present.

The preceding discussion is speci¬c to point particles in four dimensions,

but it generalizes to p-branes in D dimensions. The important point to

remember from Chapter 6 is that a charged p-brane has a (p + 1)-form

conserved current, and hence a p-form charge. To analyze such cases the

supersymmetry algebra needs to be generalized to cases appropriate to D

dimensions and p-form central charges. Calling them central is a bit of a

misnomer in this case, because for p > 0 they carry Lorentz indices and

therefore do not commute with Lorentz transformations.

One very important conclusion from the BPS bound given above is that

BPS states, which have M = |Z1 | and belong to a short multiplet, are stable.

The mass is tied to a central charge, and this relation does not change as

parameters are varied if the supersymmetry is unbroken. The only way in

which this could fail is if another representation becomes degenerate with the

BPS multiplet, so that they can pair up to give a long representation. The