Type IIA supergravity

The action of 11-dimensional supergravity is related to the actions of the

various ten-dimensional supergravity theories, which are the low-energy ef-

fective descriptions of superstring theories. The most direct connection is

between 11-dimensional supergravity and type IIA supergravity. The deep

reason is that M-theory compacti¬ed on a circle of radius R corresponds

to type √ superstring theory in ten dimensions with coupling constant

IIA

gs = R/ ± . This duality is discussed later in this chapter.4 For now, the

important consequence is that it implies that type IIA supergravity can be

obtained from 11-dimensional supergravity by dimensional reduction. Di-

mensional reduction is achieved by taking one dimension to be a circle and

only keeping the zero modes in the Fourier expansions of the various ¬elds.

This is to be contrasted with compacti¬cation, where all the modes are kept

4 In particular, it turns out that the type IIA superstring can be obtained from the M2-brane

by wrapping one dimension of the membrane on the circle to give a string in the other ten

dimensions.

308 M-theory and string duality

in the lower-dimensional theory. In fact, the type IIA supergravity action

was originally constructed by dimensional reduction. This is the easiest

method, so it is utilized in the following.

Fermionic ¬elds

As we already discussed in Chapter 5, the massless fermions of type IIA

supergravity consist of two Majorana“Weyl gravitinos of opposite chirality

and two Majorana“Weyl dilatinos of opposite chirality. These fermionic

¬elds can be obtained by taking an 11-dimensional Majorana gravitino and

dimensionally reducing it to ten dimensions. The 32-component Majorana

spinors ΨM give a pair of 16-component Majorana“Weyl spinors of oppo-

site chirality. Then the ¬rst ten components give the two ten-dimensional

gravitinos and Ψ11 gives the two ten-dimensional dilatinos. Each type IIA

dilatino has eight physical polarizations, because the Dirac equation implies

that half of the 16 components describe independent propagating modes. For

the counting to add up, it is clear that each of the gravitinos must have 56

physical degrees of freedom. These are the dimensions of irreducible repre-

sentations of Spin(8), so the discussion given here can be understood group

theoretically as the decomposition of the 128 representation of Spin(9) into

irreducible representations of the subgroup Spin(8). Altogether, there are

128 fermionic degrees of freedom, just as in 11 dimensions. This preserva-

tion of degrees of freedom is a general feature of dimensional reduction on

circles or tori.

Bosonic ¬elds

Let us now consider the dimensional reduction of the bosonic ¬elds of 11-

dimensional supergravity, the metric and the three-form. Greek letters

µ, ν, . . . refer to the ¬rst ten components of the 11-dimensional indices M, N ,

which are chosen to take the values 0, 1, . . . , 9, 11. Note that we skip the in-

dex value 10. The metric is decomposed according to

gµν + e2¦ Aµ Aν e2¦ Aµ

’2¦/3

GM N = e , (8.23)

e2¦ Aν e2¦

where all of the ¬elds depend on the ten-dimensional space-time coordinates

xµ only. The exponential factors of the scalar ¬eld ¦, which turns out to be

the dilaton, are introduced for later convenience. From the decomposition

of the 11-dimensional metric (8.23) one gets a ten-dimensional metric gµν ,

a U (1) gauge ¬eld Aµ and a scalar dilaton ¬eld ¦. Equation (8.23) can be

recast in the form

ds2 = GM N dxM dxN = e’2¦/3 gµν dxµ dxν + e4¦/3 (dx11 + Aµ dxµ )2 . (8.24)

8.1 Low-energy e¬ective actions 309

A

In terms of the elfbein EM this reduction takes the form

e’¦/3 ea 0

A µ

EM = 2¦/3 , (8.25)

2¦/3 A

e µe

where ea is the ten-dimensional zehnbein. The corresponding inverse elfbein,

µ

which is useful in the following, is given by

e¦/3 eµ 0

a

M

EA = ’2¦/3 . (8.26)

¦/3 A

’e ae

The three-form in D = 11 gives rise to a three-form and a two-form in

D = 10

(11)

A(11) = Aµνρ and Aµν11 = Bµν , (8.27)

µνρ

with the corresponding ¬eld strengths given by

(11) (11)

Fµνρ» = Fµνρ» and Fµνρ11 = Hµνρ . (8.28)

The dimensional reduction can lead to somewhat lengthy formulas due to

the nondiagonal form of the metric. A useful trick for dealing with this is

to convert ¬rst to tangent-space indices, since the reduction of the tangent-

space metric is trivial. With this motivation, let us expand

(11) (11)

MNPQ

FABCD = EA EB EC ED FM N P Q . (8.29)

There are two cases depending on whether the indices (A, B, C, D) are purely

ten-dimensional or one of them is 11-dimensional

(11)

Fabcd = e4¦/3 (Fabcd + 4A[a Hbcd] ) = e4¦/3 Fabcd ,

(8.30)

(11)

= e¦/3 Habc .

Fabc11

It follows that upon dimensional reduction the 11-dimensional ¬eld strength

is a combination of a four-form and a three-form ¬eld strength

F(4) = e4¦/3 F(4) + e¦/3 H(3) “11 , (8.31)

where “11 is the ten-dimensional chirality operator. The quantities F(4) and

H(3) are de¬ned in the same way as F(4) in Eq. (8.13). Using di¬erential-

form notation, the rescaled tensor ¬eld can be written as

F4 = dA3 + A1 § H3 . (8.32)

Notice that for the four-form F4 to be invariant under the U (1) gauge

310 M-theory and string duality

transformation δA1 = dΛ, the three-form potential should transform as

δA3 = dΛ § B. Then

δ F4 = d(dΛ § B) + dΛ § H3 = 0. (8.33)

In addition, the four-form F4 is invariant under the more obvious gauge

transformation δA3 = dΛ2 .

Coupling constants

The vacuum expectation value of exp ¦ is the type IIA superstring coupling

constant gs . From Eq. (8.24) we see that if a distance in string units is 1,

’1/3

say, then the same distance measured in 11d Planck units is gs . For small

gs , this is large. It follows that the Planck length is smaller than the string

length if gs is small. As a result,5

√

1/3

p = gs with s= ±. (8.34)

s

In ten dimensions the relation between Newton™s constant, the gravita-

tional coupling constant and the string length and coupling constant is

1

16πG10 = 2κ2 = (2π s )8 gs .

2

(8.35)