Renormalizability

By conventional power counting, e¬ective supergravity theories are non-

renormalizable. A good guide to assessing this is to examine the dimensions

of various terms in the action. The Einstein“Hilbert action, for example, in

D dimensions takes the form

√

1

’gRdD x.

S= (8.5)

16πGD

The curvature has dimensions (length)’2 , and therefore the D-dimensional

Newton constant GD must have dimension (length)D’2 . This is proportional

to the square of the gravitational coupling constant, which therefore has

negative mass dimension for D > 2. Ordinarily, barring some miracle, this

is an indication of nonrenormalizability.1 It has been shown by explicit

calculation that no such miracle occurs in the case of pure gravity in D = 4.

There is no good reason to expect miraculous cancellations in other cases

with D > 3, either, though it would be nice to prove that they don™t occur.

Nonrenormalizability is okay for theories whose only intended use is as

e¬ective actions for describing the low-energy physics of a more fundamental

theory (string theory or M-theory). The in¬nite number of higher-order

quantum corrections to these actions can be ignored for most purposes at

low energies. Some of these quantum corrections are important, however.

In fact, some of them already arose in the anomaly analysis of Chapter 5.

M-theory certainly requires an in¬nite number of higher-dimension correc-

tions to 11-dimensional supergravity. Such an expansion is unambiguously

determined by M-theory (up to ¬eld rede¬nitions) if one assumes a simple

space-time topology, such as 10,1 . In Chapter 9 it is shown that in 10,1

¡ ¡

there are R4 terms, in particular. The present chapter describes dualities

relating M-theory to type IIA and type IIB superstring theory. These have

been used to determine the precise form of the R4 corrections to D = 11

supergravity required by M-theory.

1 Actually, pure gravity for D = 3 appears to be a consistent quantum theory. However, a

graviton in three dimensions has no physical polarization states, so that theory is essentially

topological.

302 M-theory and string duality

Eleven-dimensional supergravity

The low-energy e¬ective action of M-theory, called 11-dimensional super-

gravity, is our starting point. This theory was constructed in 1978 and

studied extensively in subsequent years, but it was only in the mid-1990s

that this theory found its place on the string theory map.

In its heyday (around 1980) there were two major reasons for being skep-

tical about D = 11 supergravity. The ¬rst was its evident lack of renormal-

izability, which led to the belief that it does not approximate a well-de¬ned

quantum theory. The second was its lack of chirality, that is, its left“right

symmetry, which suggested that it could not have a vacuum with the chiral

structure required for a realistic model. Within the conventional Kaluza“

Klein framework being explored at that time, both of these objections were

justi¬ed. However, we now view D = 11 supergravity as a low-energy e¬ec-

tive description of M-theory. As such, there are good reasons to believe that

there is a well-de¬ned quantum interpretation. The situation with regard

to chirality is also changed. Among the new ingredients are the branes, the

M2-brane and the M5-brane, as well as end-of-the-world 9-branes. As was

mentioned in Chapter 5, and is discussed further in this chapter, the latter

appear in the strong-coupling description of the E8 — E8 heterotic string

theory and introduce left“right asymmetry consistent with anomaly can-

cellation requirements. There are also nonperturbative dualities, which is

discussed in this chapter, that relate M-theory to chiral superstring theories.

Moreover, it is now understood that compacti¬cation on manifolds with suit-

able singularities, which would not be well de¬ned in a pure Kaluza“Klein

supergravity context, can result in chirality in four dimensions.

Field content

Compared to the massless spectrum of the ten-dimensional superstring the-

ories, the ¬eld content of 11-dimensional supergravity is relatively simple.

First, since it contains gravity, there is a graviton, which is a symmetric

traceless tensor of SO(D ’ 2), the little group for a massless particle. It has

1 1

(D ’ 1)(D ’ 2) ’ 1 = D(D ’ 3) = 44 (8.6)

2 2

physical degrees of freedom (or polarization states). The ¬rst term counts

the number of independent components of a symmetric (D ’ 2) — (D ’ 2)

matrix and 1 is subtracted due to the constraint of tracelessness. Since this

theory contains fermions, it is necessary to use the vielbein formalism and

A

represent the graviton by a vielbein ¬eld EM . This can also be called an

elfbein ¬eld in the case of 11 dimensions, since viel is German for many, and

8.1 Low-energy e¬ective actions 303

elf is German for 11. The indices M, N, . . . are used for base-space (curved)

vectors in 11 dimensions, and the indices A, B, . . . are used for tangent-space

(¬‚at) vectors. The former transform nontrivially under general coordinate

transformations, and the latter transform nontrivially under local Lorentz

transformations.2

The gauge ¬eld for local supersymmetry is the gravitino ¬eld ΨM , which

has an implicit spinor index in addition to its explicit vector index. For

each value of M , it is a 32-component Majorana spinor. When spinors

are included, the little group becomes the covering group of SO(9), which

is Spin(9). It has a real spinor representation of dimension 16. Group

theoretically, the Spin(9) Kronecker product of a vector and a spinor is

9 — 16 = 128 + 16. The analogous construction in four dimensions gives

spin 3/2 plus spin 1/2. As Rarita and Schwinger showed in the case of a

free vector-spinor ¬eld in four dimensions, there is a local gauge invariance

of the form δΨM = ‚M µ, which ensures that the physical degrees of freedom

are pure spin 3/2. The kinetic term for a free gravitino ¬eld ΨM in any

dimension has the structure

ΨM “M N P ‚N ΨP dD x.

SΨ ∼

Due to the antisymmetry of “M N P , for δΨM = ‚M µ this is invariant up to

a total derivative.

In the case of 11 dimensions this local symmetry implies that the phys-

ical degrees of freedom correspond only to the 128. Therefore, this is the

number of physical polarization states of the gravitino in 11 dimensions. In

the interacting theory this local symmetry is identi¬ed as local supersymme-

try. This amount of supersymmetry gives 32 conserved supercharges, which

form a 32-component Majorana spinor. This is the dimension of the minimal

spinor in 11 dimensions, so there couldn™t be less supersymmetry than that

in a Lorentz-invariant vacuum. Also, if there were more supersymmetry, the

representation theory of the algebra would require the existence of massless

states with spin greater than two. It is believed to be impossible to construct

consistent interacting theories with such higher spins in Minkowski space-

time. For this reason, one does not expect to ¬nd nontrivial supersymmetric

theories for D > 11.

In order for the D = 11 supergravity theory to be supersymmetric, there

must be an equal number of physical bosonic and fermionic degrees of free-

dom. The missing bosonic degrees of freedom required for supersymmetry

2 The reader not familiar with these concepts can consult the appendix of Chapter 9 for some

basics. These also appeared in the anomaly analysis of Chapter 5.

304 M-theory and string duality

are obtained from a rank-3 antisymmetric tensor, AM N P , which can be rep-

resented as a three-form A3 . As usual for such form ¬elds, the theory has

to be invariant under the gauge transformations

A3 ’ A3 + dΛ2 , (8.7)

where Λ2 is a two-form. As is always the case for antisymmetric tensor

gauge ¬elds, including the Maxwell ¬eld, the gauge invariance ensures that

the indices for the independent physical polarizations are transverse. In the

case of a three-form in 11 dimensions this means that there are 9·8·7/3! = 84

physical degrees of freedom. Together with the graviton, this gives 44+84 =

128 propagating bosonic degrees of freedom, which matches the number of

propagating fermionic degrees of freedom of the gravitino, which is the only

fermi ¬eld in the theory.

Action

The requirement of invariance under A3 gauge transformations, together

with general coordinate invariance and local Lorentz invariance, puts strong

constraints on the form of the action. As in all supergravity theories, di-

mensional analysis determines that the number of derivatives plus half the

number of fermi ¬elds is equal to two for each term in the action. This re-

quirement reduces the arbitrariness to a few numerical coe¬cients. Finally,