Mathematically, this is just like momentum conservation at a vertex in a

Feynman diagram (in two dimensions). The junction con¬guration is stable

if the angles are chosen so that the three tensions, treated as vectors, add

to zero. It is possible to build complex string webs using such junctions.

8.3 M-theory

The term M-theory was introduced by Witten to refer to the “mysterious”

or “magical” quantum theory in 11 dimensions whose leading low-energy

e¬ective action is 11-dimensional supergravity. M-theory is not yet fully

formulated, but the evidence for its existence is very compelling. It is as

fundamental (but not more) as type IIB superstring theory, for example.

In fact, the latter is somewhat better understood precisely because it is a

string theory and therefore admits a well-de¬ned perturbation expansion.

This section describes a duality that relates M-theory compacti¬ed on a

torus to type IIB superstring theory compacti¬ed on a circle. Since this

330 M-theory and string duality

duality requires a particular geometric set-up, it only allows solutions (or

quantum vacua) of one theory to be recast in terms of the other theory for

appropriate classes of geometries.

The description of M-theory in terms of an e¬ective action is clearly not

fundamental, so string theorists are searching for alternative formulations.

One proposal for an exact nonperturbative formulation of M-theory, known

as Matrix theory, is discussed in Chapter 12. It is not the whole story,

however, since it is only applicable for a limited class of background geome-

tries. A more general approach, called AdS/CFT duality, also is discussed

in Chapter 12.

Type IIA superstring theory at strong coupling

The low-energy limit of type IIA superstring theory is type IIA supergravity,

and this supergravity theory can be obtained by dimensional reduction of

11-dimensional supergravity, as has already been discussed. However, the

correspondence between type IIA superstring theory and M-theory is much

deeper than that. So let us take a closer look at the strong-coupling limit

of the type IIA superstring theory.

D0-branes

Type IIA superstring theory has stable nonperturbative excitations, the D0-

branes, whose mass in the string frame is given by ( s gs )’1 . The claim is that

this can be interpreted from the viewpoint of M-theory compacti¬ed on a

circle as the ¬rst Kaluza“Klein excitation of the massless supergravity mul-

tiplet. The entire 256-dimensional supermultiplet is sometimes referred to

as the supergraviton. To examine this claim, let us consider 11-dimensional

supergravity (or M-theory) compacti¬ed on a circle. The mass of the super-

graviton in 11 dimensions is zero

M11 = ’pM pM = 0,

2

M = 0, 1, . . . , 9, 11. (8.103)

In ten dimensions this takes the form

M10 = ’pµ pµ = p2 ,

2

µ = 0, 1, . . . , 9. (8.104)

11

The momentum on the circle in the eleventh direction is quantized, p11 =

N/R11 , and therefore the spectrum of ten-dimensional masses is

(MN )2 = (N/R11 )2 N∈

with (8.105)

representing a tower of Kaluza“Klein excitations. These states also form

short (256-dimensional) supersymmetry multiplets, so that they are all BPS

8.3 M-theory 331

states, and carry N units of a conserved U (1) charge. For N = 1 the

correspondence with the D0-brane requires that

R11 = s gs , (8.106)

in agreement with the result presented in Section 8.1. The D0-branes are

nonperturbative excitations of the type IIA theory, since their tensions di-

verge as gs ’ 0. Therefore, this correspondence provides a test of the duality

between the type IIA theory and 11-dimensional M-theory that goes beyond

the perturbative regime.

Since R11 = s gs , the radius of the compacti¬cation is proportional to

the string coupling constant. This means that the perturbative regime of

the type IIA superstring theory in which gs ’ 0 corresponds to the limit

R11 ’ 0. Conversely, the strong-coupling limit, that is, the limit gs ’ ∞,

corresponds to decompacti¬cation of the circular eleventh dimension giving

a theory in which all ten spatial dimensions are on an equal footing. The

11-dimensional theory obtained in this way is M-theory, and the low-energy

limit of M-theory is 11-dimensional supergravity.

Turning the argument around, this is powerful evidence in support of

a nontrivial result concerning the existence of bound states of D0-branes.

The N th Kaluza“Klein excitation gives a multiplet of stable particles in ten

dimensions that have N units of charge. Therefore, they can be regarded

as bound states of N D0-branes. However, these are a very special type of

bound state, one that has zero binding energy. There is no room for any

binding energy, since these states saturate a BPS bound, which means they

are as light as they are allowed to be for a state with N units of D0-brane

charge. It also means that the mass formula in Eq. (8.105) is exact for all

values of gs . As discussed earlier, the only way in which the BPS mass

formula could be violated would be for the short supermultiplet to turn

into a long supermultiplet. However, the degrees of freedom that would be

needed for this to happen are not present in this case.

Bound states with zero binding energy are called threshold bound states,

and the question of whether or not they are stable is a very delicate matter.

From the Kaluza“Klein viewpoint it is clear that they should be stable, but

from the point of view of the dynamics of D0-branes in the type IIA theory,

it is not at all obvious. In fact, the proof is highly technical involving an

index theorem for a family of non-Fredholm operators. Moreover, the result

is speci¬c to this particular problem. There are other instances in which

coincident BPS states do not form threshold bound states. An example

that we already encountered concerns the type IIB (p, q) strings. These

strings are only stable bound states if p and q are coprime.

332 M-theory and string duality

M-branes

The BPS branes of M-theory are the M2-brane and the M5-brane. M-theory

on 11 does not contain any strings. This raises the following question:

¡

What happens to the type IIA fundamental string for large coupling, when

the theory turns into M-theory? The only plausible guess is that the type

IIA F-string is actually an M2-brane with a circular dimension wrapping

the circular eleventh dimension. Since tension is energy density, this identi-

¬cation requires that

TF1 = 2πR11 TM2 . (8.107)

This relation is satis¬ed by the tensions

1 2π

TF1 = and TM2 = , (8.108)

2π 2 (2π p )3

s

1/3

as can be veri¬ed using R11 = s gs and p = gs s . All of these relations

were presented earlier, and the proposal presented here con¬rms that they

are correct. Various other branes can be matched in a similar manner. For

example, the D4-brane is identi¬ed to be an M5-brane with one dimension

wrapped on the spatial circle.

Another interesting fact can be deduced by considering the M-theory ori-

gin of a type IIA con¬guration in which an F-string ends on a D4-brane.

In view of the above, this clearly corresponds to an M2-brane ending on

an M5-brane, where each of the M-branes is wrapped around the circular

dimension. One reason that a type IIA F-string can end on a D-brane is

that the ¬‚ux associated with the charge at the end of the string is carried

away by the one-form gauge ¬eld of the D-brane world-volume theory. That

being the case, one can ask what is the corresponding mechanism for M-

branes. The end of the M2-brane is a string inside the M5-brane. So the

world-volume theory of the M5-brane must contain a two-form gauge ¬eld

A2 to carry away the associated ¬‚ux. That is indeed the case. In fact,

the corresponding ¬eld strength F3 is self-dual, just like the ¬ve-form ¬eld

strength in type IIB supergravity.