Gravity and Strings

14.2 Quantum theories of strings 421
attached. In that case, only the U(n) vectors V IJ
µ antisymmetric in IJ survive, i.e. the
SO(n) or Sp(n) subgroups.
In general, the quotient of a theory by a discrete symmetry of the theory (such as here)
is called an orbifold by analogy with the spacetime orbifolds discussed in Section 11.6. If
in a string theory worldsheet parity is combined with a discrete symmetry of the spacetime,
then the quotient is called an orientifold [147, 148, 283, 464, 541, 542, 768, 827], 13
but since orbifolds and orientifolds are often related by dualities, it is customary to call
all of them orientifolds. In this language, closed-unoriented-string theory is an orientifold
of closed-oriented-string theory. The hypersurfaces left invariant by the discrete spacetime
symmetries are called orientifold planes and, although they are similar in other respects
to Dp-branes, they are not dynamical objects in the sense that they are attached to the
fixed points of the spacetime orbifold and cannot translate or oscillate. However, there are
dynamical fields on them. In the above case, there is no spacetime symmetry, the whole
spacetime is invariant and we can say that there is an orientifold plane of 25 spatial dimensions
(O25-plane) that fills the entire spacetime, as a D25-brane does.
There is a crucial difference between orientifolds of point-particle theories and closedstring
theories: in the latter one must include, for consistency, twisted sectors: strings that
are closed up to a symmetry operation associated with the orientifold. In general, the inclusion
of these twisted sectors makes the string theory non-singular at the orbifold fixed
points, as distinct from point-particle theories. Twisted sectors also appear in other contexts:
for instance, the winding modes that appear in toroidal compactifications (see Section 14.3)
can be seen as strings in R n closed up to an element of Ɣ n , where Ɣ n is the discrete group
used to define the torus: T n ≡ R n /Ɣ n (Ɣ n = Z n in the simplest case).
In bosonic-string theory we can add D-branes and O-planes more or less at will, because
the theory is already inconsistent due to the tachyon. In the consistent superstring theories,
D-branes and O-planes have to be introduced, paying attention to anomaly and tadpole cancelations.
These conditions are, in turn, related to the possibility of solving the equations
of motion of the effective string theory for a background that contains those objects. In
particular one has to be able to solve the harmonic equation for (p + 1)-form potentials
in compact spaces, which is possible only if the total charge associated with the potentials
vanishes. (Super-)Dp-branes and Op-planes of superstring theories are charged with
respect to (so-called) RR (p + 1)-form potentials, which we will define later, and a consistent
background will be one in which the sum of those charges vanishes. A very interesting
example, as we will see, is the construction of the type-I SO(32) superstring theory by
adding D9-branes and O9-planes to the type-IIB superstring theory [827].
There is one last consideration we must make: as we are going to see, open-string interactions
can produce closed strings. Thus, open strings are not fully consistent by themselves
and have to be combined with a closed-string sector with the same orientability. The
fields of the massless spectra of the resulting theories (without D-branes) are represented in
Table 14.2. The consistency of the interacting theory also requires the addition of twisted
sectors in orbifolds and orientifolds.
13 Forareview on orientifolds, see, for instance, [280] and also [44].