Kiefer C. Quantum gravity

298 STRING THEORY
There exist two different versions referring to gauge groups SO(32) and E 8 × E 8 ,
respectively. Anomaly-free chiral models for particle physics can, thus, be constructed
from string theory for these gauge groups. This has raised the hope that
the Standard Model of strong and electroweak interactions can be derived from
string theory—a hope, however, which up to now has not been realized.
To summarize, one has found (the weak-field limit of) five consistent string
theories in D = 10 dimensions. To find the theory in four dimensions, one has
to invoke a compactification procedure. Since no principle has yet been found to
fix this, there are a plenty of consistent string theories in four dimensions and it
is not clear which one to choose.
This is, however, not yet the end of the story. Type IIA theory also contains
D0-branes (‘particles’); cf. Witten (1995). If one has n such D0-branes, their
mass M is given by
M =
n
g √ . (9.68)
α ′
In the perturbative regime g ≪ 1, this state is very heavy, while in the strongcoupling
regime g →∞, it becomes lighter than any perturbative excitation.
The mass spectrum (9.68) resembles a Kaluza–Klein spectrum; cf. the beginning
of Section 9.2.6. It thus signals the presence of an 11th dimension with radius
R 11 = g √ α ′ . (9.69)
The 11th dimension cannot be seen in string perturbation theory, which is a
perturbation theory for small g. SinceD = 11 is the maximal dimension in which
SUSY can exist, this suggests a connection with 11-dimensional SUGRA. It is
generally believed that the five string theories are the perturbative limits of one
fundamental theory of which 11-dimensional SUGRA is a low-energy limit. This
fundamental theory, about which little is known, is called M-theory. A particular
proposal of this theory is ‘matrix theory’ which employs only a finite number of
degrees of freedom connected with a system of D0-branes (Banks et al. 1997).
Its fundamental scale is the 11-dimensional Planck length. The understanding
of M-theory is indeed very limited. It is, for example, not yet possible to give
a full non-perturbative calculation of graviton–graviton scattering, one of the
important processes in quantum gravity (see Chapter 2).
In Section 9.2.3, we have discussed the notion of T-duality, which connects
descriptions of small and large radii. There is a second important notion of
duality called ‘S-duality’, which relates the five consistent superstring theories
to each other. Thereby the weak-coupling sector (g ≪ 1) of one theory can be
connected to the strong-coupling sector (g ≫ 1) of another (or the same) theory.
9.2.5 Black-hole entropy
In Section 7.3, we have reviewed attempts to calculate the Bekenstein–Hawking
entropy (7.17) by counting microscopic degrees of freedom in canonical quantum
gravity; see also the remarks in Section 8.1.3 on the situation in (2+1)-
dimensional gravity. What can string theory say about this issue? It turns out