Gravity and Strings

14.2 Quantum theories of strings 425
6. As we have mentioned, the absence of anomalies and tadpoles is related to the stability
of the system of Dp-branes and Op-planes. In particular, the system should be
able to solve the equations of motion of the string effective theory which we are going
to study in the next few chapters. The equations of motion for (p + 1)-forms are
generalizations of the harmonic equation which, in compact spaces, can be solved
only if the total charge is zero. 15
These rules have been used extensively for building new string theories. The simplest
construction leads to the type-I SO(32) theory starting from type IIB: on introducing an
O9-plane (i.e. taking the quotient of the type-IIB theory by ), consistency requires the
addition of 32 D9-branes in order to obtain zero total RR charge, which results in the
introduction of an open, unoriented-string sector with gauge group SO(32).
14.2.4 String interactions
Strings interact by joining and splitting. It is then easy to understand that open strings can
interact to give closed strings and that consistency (unitarity) requires a closed-string sector
in open-string theories.
String amplitudes are defined as path integrals over all embeddings X µ and all worldsheet
metrics γij with given boundaries and boundary data that determine the string states
that are scattered. The boundary data are included as vertex operators in the path integral.
Without vertex operators, we have vacuum amplitudes, given by the path integral
Z = DXDγ e −SP−SEuler , (14.55)
where SP is the Euclidean Polyakov integral Eq. (14.5) and SEuler is the topological term
Eq. (14.14). For closed strings we will restrict ourselves to (just oriented or oriented plus
unoriented) compact surfaces. For open strings we will add surfaces with boundaries.
The sum over metrics can be decomposed into a sum of path integrals over worldsheets
with given topologies. The topology of two-dimensional surfaces can be characterized completely
by the numbers g, b, and c,combined into the Euler characteristic χ, which is given
by the topological term Eq. (14.14) as explained on page 412. The result takes the form
Z =
(e φ0
−χ(t)
) DXDγ e −SPt , (14.56)
t
where t stands for given topologies and {t} is the space of surfaces with topology t.Now,
each topology can be associated with a loop order, given precisely by −χ(t), and the above
sum can be understood as a perturbative series expansion in which e φ0 plays the role of the
string coupling constant g:
{t }
g ≡ e φ0. (14.57)
In Section 15.1 we will see that φ0 is the vacuum expectation value of the dilaton field.
15 The lines of force of the field can only go to sources or to infinity. In compact spacetimes, they have to start
and end on sources and the total charge has to be zero.