Kiefer C. Quantum gravity

QUANTUM-GRAVITATIONAL ASPECTS 295
cf. the remarks before (9.2)). Since, therefore, X D (π) andX D (0) differ by a
multiple of the internal circumference 2πR D of the dual space, the endpoints
must lie on a (D − 1)-dimensional hypersurface. Consideration of several open
strings reveals that it is actually the same hypersurface. This hypersurface is
called a D-brane where ‘D’ refers to the Dirichlet condition holding normal to
the brane. Sometimes one refers to it more precisely as the ‘Dp-brane’, where p
is the number of space dimensions.
AD-braneisadynamical object since momentum can leak out of the string
and is absorbed by the brane (this cannot happen with a Neumann boundary
condition which still holds in the directions tangential to the plane). A D-brane is
a soliton of string theory and can be described by an action that resembles an action
proposed long ago by Born and Infeld to describe non-linear electrodynamics
(it was then meant as a candidate for a modification of linear electrodynamics at
short distances). A D-brane can carry generalized electric and magnetic charges.
The above discussion can be extended to the presence of gauge fields. The
reason behind this is that open strings allow additional degrees of freedom called
‘Chan–Paton factors’. These are ‘charges’ i and j (i, j =1,...,n) which reside at
the endpoints of the string (historically one was thinking about quark–antiquark
pairs). One can introduce a U(n) symmetry acting on (and only on) these charges.
One can get from this the concept of U(n) gauge bosons living on the branes
(one can have n branes at different positions).
Interestingly, n coinciding D-branes give rise to ‘n × n’-matrices for the embedding
variables X µ and the gauge fields A a . One thus arrives at space–time
coordinates that do not commute, giving rise to the notion of non-commutative
space–time. It has been argued that the D-brane action corresponds to a Yang–
Mills action on a non-commutative worldvolume. Details are reviewed, for example,
in Douglas and Nekrasov (2002).
The concept of D-branes is especially interesting concerning gravitational
aspects. First, it plays a crucial role for the derivation of the black-hole entropy
from counting microscopic degrees of freedom (Section 9.2.5). The second point
has to do with the fact that these branes allow one to localize gauge and matter
fields on the branes, whereas the gravitational field can propagate through the
full space–time. This gives rise to a number of interesting features discussed in
the context of ‘brane worlds’; cf. Section 9.2.6.
9.2.4 Superstrings
So far, we have not yet included fermions, which are necessary for a realistic
description of the world. Fermions are implemented by the introduction of SUSY,
which we have already discussed in Sections 2.3 and 5.3.6. In contrast to the
discussion there, we shall here introduce SUSY on the worldsheet, not on space–
time. This will help us to get rid of problems of the bosonic string, such as the
presence of tachyons. A string with SUSY is called ‘superstring’. Worldsheet
SUSY will be only indirectly related to space–time SUSY. We shall be brief in
our discussion and refer the reader to, for example, Polchinski (1998b) formore