in the Higgs mechanism, where a massless vector (a short representation of

the Lorentz group) joins up with a scalar to give a massive vector (a long

representation) as a parameter in the Higgs potential is varied. The thing

that is di¬erent about supersymmetric examples is that short multiplets can

be massive. In any case, the conclusion is that so long as such a joining of

multiplets does not happen, it is possible to follow BPS states from weak

coupling to strong coupling with precise control. This is very important for

testing conjectures about the behavior of string theories at strong coupling,

as we shall see in this chapter.

EXERCISES

EXERCISE 8.1

The N = 1 supersymmetry algebra in four dimensions does not have a

central extension. The explicit form of this algebra, with the supercharges

M-theory and string duality 299

expressed as two-component Weyl spinors Q± and Qβ = Q† , is

™ β

{Q± , Qβ } = 2σ µ ™ Pµ , {Q± , Qβ } = {Q± , Qβ } = 0.

and

™ ™

™

±β

Determine the irreducible massive representations of this algebra.

SOLUTION

As in the text, for massive states we can work in the rest frame, where the

momentum vector is Pµ = (’M, 0, 0, 0). Then the algebra becomes

10

{Q± , Qβ } = 2M δ±β = 2M .

™ ™

01

This algebra is a Cli¬ord algebra, so it is convenient to rescale the operators

to obtain a standard form for the algebra

1 1

b† = √

b± = √ Q± and Q± .

™

±

2M 2M

The supersymmetry algebra then becomes

{b± , b† } = δ±β , {b± , bβ } = {b† , b† } = 0.

±β

β

As a result, b± and b† act as fermionic lowering and raising operators, and we

±

obtain all the states in the supermultiplet by acting with raising operators b†

±

on the Fock-space ground state |„¦ , which satis¬es the condition b± |„¦ = 0.

1

Then, if |„¦ represents a state of spin j, a state of spin j ± 2 is created

by acting with the fermionic operators b† |„¦ . If the ground state |„¦ has

±

†

spin 0 (a boson), then b± |„¦ represent the two states of a spin 1/2 fermion.

Moreover b† b† |„¦ gives a second spin 0 state. In general, for a ground state

12

of spin j > 0 and multiplicity 2j + 1, this construction gives the 4(2j + 1)

states of a massive representation of N = 1 supersymmetry in D = 4 with

spins j ’ 1/2, j, j, j + 1/2. 2

EXERCISE 8.2

Determine the multiplet structure for massive states of N = 2 supersymme-

try in four dimensions in the presence of the central charge. In particular

derive the form of the short and long multiplets.

SOLUTION

For N = 2 supersymmetry the central charge is Z IJ = ZµIJ . For simplicity,

let us assume that Z is real and nonnegative. Using this form of the central

300 M-theory and string duality

charge, the supersymmetry algebra in the rest frame can be written in the

form

J

{QI , Qβ } = 2M δ±β δ IJ ,

™ ™

±

{QI , QJ } = 2Zµ±β µIJ ,

± β

I J

{Q± , Qβ } = 2Zµ±β µIJ ,

™ ™™

™

where I, J = 1, 2. We rearrange these generators and de¬ne

2 1

b± = Q1 ± µ±β Qβ (b± )† = Q± ± µ±β Q2 .

and

± ± ± β

Note that this construction identi¬es dotted and undotted indices. This

is sensible because a massive particle at rest breaks the SL(2, ) Lorentz

£

group to the SU (2) rotational subgroup, so that the 2 and ¯ representa-

2

tions become equivalent. It is then easy to verify that the only nonzero

anticommutators of these generators are

{b’ , (b’ )† } = 4δ±β (M ’ Z).

{b+ , (b+ )† } = 4δ±β (M + Z) and

± ±

β β

These anticommutation relations give the BPS bound for N = 2 theories,

which takes the form

M ≥ Z.

If this bound is not saturated, we can act with (b± )† on a spin j ground

±

state |„¦ to create the 16(2j + 1) states of a long supermultiplet. However,

if the BPS bound is saturated, that is, if M = Z, then the physical states in

the supermultiplet are created by acting only with (b+ )† . This reduces the

±

number of states to 4(2j + 1) and creates a short supermultiplet. The case

j = 0 gives a half hypermultiplet. Such a multiplet is always paired with its

TCP conjugate to give a hypermultiplet with four scalars and two spinors.

The case j = 1/2 gives a vector multiplet. 2

8.1 Low-energy e¬ective actions

Previous chapters have described how the spectrum of states of the various

superstring theories behaves in the weak-coupling limit. The masses of all

states other than the massless ones become very large for ± ’ 0, which

corresponds to large string tension. Equivalently, at least in a Minkowski

space background where there is no other scale, this corresponds to the

low-energy limit, since the only dimensionless parameter is ± E 2 . In the

8.1 Low-energy e¬ective actions 301

low-energy limit, it is a good approximation to replace string theory by a

supergravity theory describing the interactions of the massless modes only, as

the massive modes are too heavy to be observed. This section describes the

supergravity theories arising in the low-energy limit of string theory. These

theories are not fundamental, but they do capture some of the important

features of the more fundamental string theories.