generates the entire U-duality group in every dimension. It is a property

of quantum M-theory that goes beyond what can be understood from the

e¬ective 11-dimensional supergravity theory, which is geometrical.

EXERCISES

EXERCISE 8.7

Verify Eqs (8.131) and (8.132).

8.4 M-theory dualities 349

SOLUTION

Since the M-theory metric and the type IIB metric are related by

g (M) = β 2 g (B) ,

masses are related according to

M11 = βMB .

Matching the mass of an M2-brane wrapped on the torus with a Kaluza“

Klein excitation on the type IIB circle therefore gives

1

AM TM2 = β .

RB

Similarly, using Eq. (8.128) for the mass of a Kaluza“Klein excitation on

the torus, and equating it to the mass of a wrapped type IIB string gives

1

= β(2πRB TF1 ).

R11 Im„M

Multiplying these equations together, using AM = (2πR11 )2 Im„M , gives

RB AM TM2 2πR11 TM2

β2 = = ,

2πRB TF1 R11 Im„M TF1

which is Eq. (8.131). Taking the quotient of the same two equations, using

gs = (Im„M )’1 , gives

2

gs

= TM2 (2πR11 )3 ,

2T

RB F1

which is Eq. (8.132). 2

EXERCISE 8.8

Identifying type IIB superstring theory compacti¬ed on a circle and M-

theory compacti¬ed on a torus, match the tensions of the nine-dimensional

4-branes.

SOLUTION

A (p, q) type IIB 5-brane wrapped on the circle is identi¬ed with an M5-

brane wrapping a geodesic (p, q) cycle of the torus. Equating the resulting

tensions gives

2πRB β 5 T(p,q) = L(p,q) TM5 ,

where L(p,q) = 2πR11 |p ’ q„M |. We can check that the resulting D5-brane

350 M-theory and string duality

tension in ten dimensions agrees with the result quoted in Chapter 6. Indeed,

setting p = 1 and q = 0, we obtain

3

TF1

R11 ’5 1

TD5 = β TM5 = = ,

(2π)2 gs (2π)5 6 gs

RB s

which is the TD5 derived in Chapter 6. Therefore,

T(p,q) = |p ’ q„M |TD5 .

In particular, the NS5-brane tension is obtained by setting q = 1 and p = 0

TNS5 = |„M |TD5 .

The standard result is obtained by setting „M = i/gs . 2

EXERCISE 8.9

Verify that the three groups (8.145) are maximally noncompact.

SOLUTION

The group SO(5, 5) has dimension equal to 45, just like its compact form

SO(10). Its maximal compact subgroup is SO(5)—SO(5), which has dimen-

sion equal to 20. Thus, there are 25 noncompact generators and 20 compact

generators. Since the rank of SO(5, 5), which is ¬ve, agrees with 25 ’ 20,

it is maximally noncompact. In the case of SL(5, ), which is a noncom-

¡

pact form of SU (5), the rank is four and the dimension is 24. The maximal

compact subgroup is SO(5), which has dimension equal to ten. Thus there

are 14 noncompact generators and ten compact generators, and once again

the di¬erence is equal to the rank. This reasoning generalizes to SL(n, ),

¡

which has (n ’ 1)(n + 2)/2 noncompact generators, (n ’ 1)n/2 compact

generators and rank n ’ 1. The group SL(3, ) — SL(2, ) is maximally

¡ ¡

noncompact, because each of its factors is. 2

EXERCISE 8.10

Find a physical interpretation of Eqs (8.159) and (8.160).

SOLUTION

Equation (8.159) implies that

1

’ (2πR2 )(2πR3 )TM2 .

R1

Thus it interchanges a Kaluza“Klein excitation of the ¬rst circle with an

Homework Problems 351

M2-brane wrapped on the second and third circles. The circles can be

permuted, so it follows that these six 0-brane excitations belong to the (3, 2)

representation of the SL(3, ) — SL(2, ) U-duality group.

Equation (8.160) implies that

TM2 ’ (2πR1 )(2πR2 )(2πR3 )TM5 .

Therefore, it interchanges an unwrapped M2-brane with an M5-brane wrap-

ped on the T 3 . Thus these two 2-branes (from the eight-dimensional view-

point) belong to the (1, 2) representation of the U-duality group. 2

HOMEWORK PROBLEMS

PROBLEM 8.1

Derive the bosonic equations of motion of 11-dimensional supergravity.

PROBLEM 8.2

Show that a particular solution of the bosonic equations of motion of 11-

dimensional supergravity, called the Freund“Rubin solution, is given by a

product space-time geometry AdS4 — S 7 with

F4 = M µ 4 ,

where µ4 is the volume form on AdS4 , and M is a free parameter with the

dimensions of mass.16 AdS4 denotes four-dimensional anti-de Sitter space,

which is a maximally symmetric space of negative curvature, with Ricci

tensor

Rµν = ’(M4 )2 gµν µ, ν = 0, 1, 2, 3.

The seven-sphere has Ricci tensor

Rij = (M7 )2 gij i, j = 4, 5, . . . , 10.