2π

Dimensional reduction on a circle of radius R11 gives a relation between

Newton™s constant in ten and 11 dimensions

G11 = 2πR11 G10 . (8.36)

Using Eqs (8.9) and (8.34), one deduces that the radius of the circle is

2/3

R11 = gs = gs s . (8.37)

p

These formulas are con¬rmed again later in this chapter when the type IIA

D0-brane is identi¬ed with the ¬rst Kaluza“Klein excitation on the circle.

Let us also de¬ne

1

2κ2 = (2π s )8 , (8.38)

2π

which agrees with 2κ2 up to a factor of gs , that is, κ2 = κ2 gs .

2 2

10 10

√

5 Chapter 2 introduced a string length scale ls = 2± , which has been used until now. Here it is

√

convenient to introduce a string length scale s = ± , which is used throughout this chapter.

Note the change of font. Both conventions are used in the literature, and there is little to be

gained from eliminating one of them.

8.1 Low-energy e¬ective actions 311

Action

The bosonic action in the string frame for the D = 10 type IIA supergravity

theory is obtained from the bosonic D = 11 action once the integration over

the compact coordinate is carried out. The result contains three distinct

types of terms

S = SNS + SR + SCS . (8.39)

The ¬rst term is

√

1 1

d10 x ’g e’2¦ R + 4‚µ ¦‚ µ ¦ ’ |H3 |2 .

SNS = 2 (8.40)

2κ 2

Note that the coe¬cient is 1/2κ2 , which does not contain any powers of the

string coupling constant gs . This string-frame action is characterized by the

exponential dilaton dependence in front of the curvature scalar. By a Weyl

rescaling of the metric, this action can be transformed to the Einstein frame

in which the Einstein term has the conventional form. This is a homework

problem.

The remaining two terms in the action S involve the R“R ¬elds and are

given by

√

1

SR = ’ 2 d10 x ’g |F2 |2 + |F4 |2 , (8.41)

4κ

1

SCS = ’ B2 § F4 § F4 . (8.42)

4κ2

As a side remark, let us point out the following: a general rule, discussed in

Chapter 3, is that a world sheet of Euler characteristic χ gives a contribution

with a dilaton dependence exp(χ¦), which leads to the correct dependence

on the string coupling constant. All terms in the classical action Eq. (8.39)

correspond to a spherical world sheet with χ = ’2, because they describe

the leading order of the expansion in gs . Notice, however, that the terms

SR and SCS , which involve R“R ¬elds, do not contain the expected factor

of e’2¦ . This is only a consequence of the way the R“R ¬elds have been

de¬ned. One could rescale C1 and F2 by C1 = e’¦ C1 and F2 = e’¦ F2 ,

where F2 = dC1 ’ d¦ § C1 and make analogous rede¬nitions for C3 and

F4 . Then the factor of e’2¦ would appear in all terms. However, this ¬eld

rede¬nition is not usually made, so the action that is displayed is in the form

that is most commonly found in the literature.

Supersymmetry transformations

Let us now examine the supersymmetry transformations of the fermi ¬elds

to leading order in these ¬elds. We ¬rst rewrite the gravitino variation in

312 M-theory and string duality

Eq. (8.12) in the form

1 1

µ

δΨA = EA ‚µ µ + ωABC “BC µ + 3F(4) “A ’ “A F(4) µ, (8.43)

4 24

where we are using 11-dimensional tangent-space indices. To interpret the

previous expression in terms of ten-dimensional quantities, we need to work

out the various pieces of the spin connection, which (to avoid confusion) is

(11)

now denoted ωABC . Using Eq. (8.19), one ¬nds that

2

(11)

ωaBC “BC = e¦/3 (ωabc “bc ’ “a µ ‚µ ¦) + e4¦/3 Fab “b “11 (8.44)

3

and

1 4

(11)

ω11BC “BC = ’ e4¦/3 Fbc “bc ’ e¦/3 “µ “11 ‚µ ¦. (8.45)

2 3

Using these equations

1 1 1 1

e’¦/3 δΨ11 = ’ e¦ F(2) µ ’ ‚µ ¦“µ “11 µ + e¦ F(4) “11 µ + H(3) µ (8.46)

4 3 12 6

and

1 1

e’¦/3 δΨa = eµ µ µ ’ “a µ ‚µ ¦µ + e¦ Fab “b “11 µ

a

6 4

1¦ 1

e (3F(4) “a ’ “a F(4) )µ ’ (3H(3) “a + “a H(3) )“11 µ.

+ (8.47)

24 24

To obtain the supersymmetry transformations in the desired form, we

de¬ne new spinors as follows:

˜

» = e’¦/6 Ψ11 , (8.48)

1

Ψµ = e’¦/6 (Ψµ + “µ “11 Ψ11 ) (8.49)

2

and µ = exp(¦/6)µ. The ¬nal expressions for the supersymmetry transfor-

˜

mations then become6

1 1 1 1

’ “µ ‚µ ¦“11 + H(3) ’ e¦ F(2) + e¦ F(4) “11 µ

δ» = (8.50)

3 6 4 12

and

1 1 1

’ H(3) “11 ’ e¦ Fνρ “µ νρ “11 + e¦ F(4) “µ µ.

δΨµ = (8.51)

µ µ

4 8 8

The second term in δΨµ has an interpretation as torsion.7 Because of the “11

6 In order to make the equations less cluttered, we have removed the tildes from the fermionic

¬elds and µ.

7 Torsion is de¬ned in the appendix of Chapter 9.

8.1 Low-energy e¬ective actions 313

1

factor, the torsion has opposite sign for the opposite chiralities 2 (1±“11 )Ψµ .

The spinors », Ψµ and µ are each Majorana spinors. As such they could

be decomposed into a pair of Majorana“Weyl spinors of opposite chirality,

though there is no advantage in doing so. Therefore, they describe two

dilatinos, two gravitinos and N = 2 supersymmetry in ten dimensions.

Type IIB supergravity

Unlike type IIA supergravity, the type IIB theory cannot be obtained by

reduction from 11-dimensional supergravity. The guiding principles to con-

struct this theory come from supersymmetry as well as gauge invariance.

One challenging feature of the type IIB theory is that the self-dual ¬ve-form

¬eld strength introduces an obstruction to formulating the action in a man-

ifestly covariant form. One strategy for dealing with this is to focus on the

¬eld equations instead, since they can be written covariantly. Alternatively,

one can write an action that needs to be supplemented by a self-duality

constraint.