String Theory and M-Theory

3.7 Witten’s open-string field theory 105
for the free theory is QBA = 0, which is invariant under the abelian gauge
transformation δA = QBΛ, since Q2 B = 0. Once one requires that A be
restricted to contain ghost number −1/2 fields, this precisely reproduces
the known spectrum of the bosonic string. As was explained earlier, the
physical states of the free theory are in one-to-one correspondence with
BRST cohomology classes of ghost number −1/2.
The cubic string interaction is depicted in part (b) of Fig. 3.8. Two of
the segment identifications are consequences of the ∗ products in A ∗ A ∗ A,
and the third is a consequence of the integration. Altogether, this gives an
expression that is symmetric in the three strings.
As was explained earlier, in string theory one is only interested in equivalence
classes of metrics that are related by conformal mappings. It is always
possible to find representatives of each equivalence class in which the metric
is flat everywhere except at isolated points where the curvature is infinite.
Such a metric describes a surface with conical singularities, which is not a
manifold in the usual sense. In fact, it is an example of a class of surfaces
called orbifolds. The string field theory construction of the amplitude automatically
chooses a particular metric, which is of this type. The conical
singularities occur at the string midpoints in the interaction. They have
the property that a small circle of radius r about this point has circumference
3πr. This is exactly what is required so that the Riemann surfaces
constructed by gluing vertices and propagators have the correct integrated
curvature, as required by Euler’s theorem.
Witten’s string field theory seems to be as simple and beautiful as one
could hope for, though there are subtleties in defining it precisely that have
been glossed over in the brief presentation given here. For the bosonic string
theory, it does allow a computation of all processes with only open-string
external lines to all orders in perturbation theory (at least in principle). The
extension to open superstrings is much harder and has not been completed
yet. It has been proved that the various Feynman diagrams generated by
this field theory piece together so as to cover the relevant Riemann surface
moduli spaces exactly once. In particular, this means that the contributions
of closed strings in the interior of diagrams is properly taken into account.
Moreover, the fact that this is a field-theoretic formulation means that it
can be used to define amplitudes with off-shell open strings, which are otherwise
difficult to define in string theory. This off-shell property has been
successfully exploited in nonperturbative studies of tachyon condensation.
However, since this approach is based on open-string fields, it is not applicable
to theories that only have closed strings. Corresponding constructions
for closed-string theories (mostly due to Zwiebach) are more complicated.