Kiefer C. Quantum gravity

QUANTUM-GRAVITATIONAL ASPECTS 291
action contains the gravitational constant G D , the comparison of the amplitudes
yields a connection between both; see, for example, Veneziano (1993),
G D ∼ g 2 l D−2
s , (9.42)
where g is again the string-coupling constant (here we do not distinguish between
open and closed strings and write for simplicity just g for the string coupling).
An analogous relation holds between gauge couplings for grand unified theories
and the string length.
Since we do not live in 26 dimensions, a connection must be made to the
four-dimensional world. This is usually done through compactification of the additional
dimensions which have the form of ‘Calabi–Yau manifolds’; cf. Candelas
et al. (1985). In this way, one obtains a relation between the four-dimensional
gravitational constant and the string length,
G ∼ g 2 l 2 s , (9.43)
in which the details of the compactification enter into geometric factors. Ideally,
one would like to recover in this way other parameters such as particle masses
or the number of families from the details of compactification. One is, however,
still far away from this goal.
The finiteness of the string length l s leads to an automatic cutoff at high
momenta. It thus seems impossible to resolve arbitrarily small distances in an
operational sense. In fact, one can derive from gedanken experiments of scattering
situations, a generalized uncertainty relation of the form
∆x >
∆p + l2 s
∆p ; (9.44)
cf. Veneziano (1993) and the references therein. This seems to match the idea
of a minimal length which we have also encountered in the canonical approach
(Section 6.2), although, for example, D-branes (Section 9.2.3) can probe smaller
scales.
How many fundamental constants appear in string theory? This has been
a matter of some debate; see Duff et al. (2002). We adopt here the standpoint
already taken in Chapter 1 that three dimensionful constants are needed, which
can be taken to be c, , andl s .
9.2.3 T-duality and branes
In this subsection, we shall introduce the concept of T-duality, from which one
is led in a natural way to the concept of D-branes. The ‘T’ arises from the fact
that one assumes here that the higher dimensions are compactified on tori.
The classical solutions for the closed string are given in Section 9.1. In order
to introduce the concept of duality, we take the left- and rightmoving modes of
the closed string as independent, that is, we write