String Theory and M-Theory

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 effective 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 geometries.
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 D0branes,
whose mass in the string frame is given by (ℓsgs) −1 . The claim is that
this can be interpreted from the viewpoint of M-theory compactified on a
circle as the first Kaluza–Klein excitation of the massless supergravity multiplet.
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) compactified on a circle. The mass of the supergraviton
in 11 dimensions is zero
M 2 11 = −pM p M = 0, M = 0, 1, . . . , 9, 11. (8.103)
In ten dimensions this takes the form
M 2 10 = −pµp µ = p 2 11, µ = 0, 1, . . . , 9. (8.104)
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
with N ∈
(8.105)
representing a tower of Kaluza–Klein excitations. These states also form
short (256-dimensional) supersymmetry multiplets, so that they are all BPS