ory in D = 11 (up to normalization conventions). In fact, it is so strongly

constrained that its existence appears quite miraculous.

The bosonic part of the 11-dimensional supergravity action is

√ 1 1

2κ2 S = d11 x ’G R ’ |F4 |2 ’ A3 § F 4 § F 4 , (8.8)

11

2 6

where R is the scalar curvature, F4 = dA3 is the ¬eld strength associated

with the potential A3 , and κ11 denotes the 11-dimensional gravitational cou-

pling constant. The relation between the 11-dimensional Newton™s constant

G11 , the gravitational constant κ11 and the 11-dimensional Planck length p

is3

1

16πG11 = 2κ2 = (2π p )9 . (8.9)

11

2π

The last term in Eq. (8.8), which has a Chern“Simons structure, is inde-

pendent of the elfbein (or the metric). The ¬rst term does depend on the

elfbein, but only in the metric combination

AB

GM N = ·AB EM EN . (8.10)

3 The coe¬cients in these relations are the most commonly used conventions.

8.1 Low-energy e¬ective actions 305

The quantity |F4 |2 is de¬ned by the general rule

1 M 1 N1 M 2 N2

|Fn |2 = · · · GMn Nn FM1 M2 ···Mn FN1 N2 ···Nn .

G G (8.11)

n!

Supersymmetry transformations

The complete action of 11-dimensional supergravity is invariant under local

supersymmetry transformations under which the ¬elds transform according

to

A = µ“A ΨM ,

δEM ¯

= ’3¯“[M N ΨP ],

δAM N P µ (8.12)

(4)

1

“M F(4) ’ 3FM

δΨM = Mµ + µ.

12

Here we have introduced the de¬nitions

1

F(4) = F M N P Q “M N P Q (8.13)

4!

and

1 1

(4)

[“M , F(4) ] = FM N P Q “N P Q .

FM = (8.14)

2 3!

Straightforward generalizations of this notation are used in the following.

The formula for δΨM displays the terms that are of leading order in fermi

¬elds. Additional terms of the form (fermi)2 µ have been dropped. The Dirac

matrices satisfy

A

“M = EM “A , (8.15)

where “A are the numerical (coordinate-independent) matrices that obey

the ¬‚at-space Dirac algebra. Also, the square brackets represent antisym-

metrization of the indices with unit weight. For example,

1

“[M N ΨP ] = (“M N ΨP + “N P ΨM + “P M ΨN ). (8.16)

3

Another convenient notation that has been used here is

“M1 M2 ···Mn = “[M1 “M2 · · · “Mn ] . (8.17)

The covariant derivative that appears in Eq. (8.12) involves the spin con-

nection ω and is given by

1

= ‚M µ + ωM AB “AB µ.

Mµ (8.18)

4

306 M-theory and string duality

The spin connection can be expressed in terms of the elfbein by

1

ωM AB = (’„¦M AB + „¦ABM ’ „¦BM A ), (8.19)

2

where

„¦M N A = 2‚[N EM ] .

A

(8.20)

In fact, these relations are valid in any dimension. Depending on conven-

tions, the spin connection may also contain terms that are quadratic in fermi

¬elds. Such terms are neglected here, since they are not relevant to the issues

that we discuss.

Supersymmetric solutions

One might wonder why the supersymmetry transformations have been pre-

sented without also presenting the fermionic terms in the action. After all,

it is the complete action including the fermionic terms that is supersym-

metric. The justi¬cation is that one of the main uses of this action, and

others like it, is to construct classical solutions. For this purpose, only the

bosonic terms in the action are required, since a classical solution always

has vanishing fermionic ¬elds.

One is also interested in knowing how many of the supersymmetries sur-

vive as vacuum symmetries of the solution. Given a supersymmetric solu-

tion, there exist spinors, called Killing spinors, that characterize the super-

symmetries of the solution. The concept is similar to that of Killing vectors,

which characterize bosonic symmetries. Killing vectors are vectors that ap-

pear as parameters of in¬nitesimal general coordinate transformations under

which the ¬elds are invariant for a speci¬c solution. In analogous fashion,

Killing spinors are spinors that parametrize in¬nitesimal supersymmetry

transformations under which the ¬elds are invariant for a speci¬c ¬eld con-

¬guration. Since the supersymmetry variations of the bosonic ¬elds always

contain one or more fermionic ¬elds, which vanish classically, these variations

are guaranteed to vanish. Thus, in exploring supersymmetry of solutions,

the terms of interest are the variations of the fermionic ¬elds that do not

contain any fermionic ¬elds. In the case at hand this means that Killing

spinors µ are given by solutions of the equation

1 (4)

“M F(4) ’ 3FM µ = 0,

δΨM = Mµ + (8.21)

12

and the bosonic terms that have been included in Eq. (8.12) determine the

possible supersymmetric solutions.

8.1 Low-energy e¬ective actions 307

M-branes

An important feature of M-theory (and 11-dimensional supergravity) is the

presence of the three-form gauge ¬eld A3 . As has been explained in Chap-

ter 6, such ¬elds couple to branes, which in turn are sources for the gauge

¬eld. In this case (n = 3 and D = 11) the three-form can couple electrically

to a two-brane, called the M2-brane, and magnetically to a ¬ve-brane, called

the M5-brane. If the tensions saturate a BPS bound (as they do), these are

stable supersymmetric branes whose tensions can be computed exactly. By

focusing attention on BPS M-branes, it is possible to learn various facts

about M-theory that go beyond the low-energy e¬ective-action expansion.

In fact, we will even discover an M-theory version of T-duality that shows

the limitations of a geometrical description.

The only scale in M-theory is the 11-dimensional Planck length p . There-

fore, the M-brane tensions can be determined, up to numerical factors, by

dimensional analysis. The exact results, which are con¬rmed by duality ar-

guments relating M-branes to branes in type II superstring theories, turn

out to be

TM2 = 2π(2π p )’3 TM5 = 2π(2π p )’6 .

and (8.22)

As is the case with all BPS branes, an M-brane can be excited so that it is

no longer BPS, but then it would be unstable and radiate until reaching the

minimal BPS energy density in (8.22).