This product structure is then reflected in the amplitudes.
Indeed, the celebrated Koba-Nielsen form of string
amplitudes [88], which may be obtained by evaluating correlations of the vertex
operators, factorize at the level of the integrands before world
sheet integrations are performed. Amazingly, Kawai, Lewellen, and
Tye were able to demonstrate a much stronger factorization:
Complete closed string amplitudes factorize into products of open
string amplitudes, even
after
integration over the world sheet variables. (A description of
string theory scattering amplitudes and the history of their
construction may be found in standard books on string
theory [70,
110,
111].)
As a simple example of the factorization property of string theory amplitudes, the four-point partial amplitude of open superstring theory for scattering any of the massless modes is given by
where
is the open string Regge slope proportional to the inverse
string tension,
g
is the gauge theory coupling, and
K
is a gauge invariant kinematic coefficient depending on the
momenta
. Explicit forms of
K
may be found in Ref. [70
]. (The metric is taken here to have signature (+, -, -, -).) In
this and subsequent expressions,
,
, and
. The indices can be either vector, spinor or group theory
indices and the
can be vector polarizations, spinors, or group theory matrices,
depending on the particle type. These amplitudes are the open
string partial amplitudes before they are dressed with
Chan-Paton [106
] group theory factors and summed over non-cyclic permutations to
form complete amplitudes.(Any group theory indices in Eq. (6
) are associated with string world sheet charges arising from
possible compactifications.) For the case of a vector,
is the usual polarization vector. Similarly, the four-point
amplitudes corresponding to a heterotic closed superstring [75,
76] are
where
is the open string Regge slope or equivalently twice the close
string one. Up to prefactors, the replacements
and substituting
, the closed string amplitude (7
) is a product of the open string partial amplitudes (6
). For the case of external gravitons the
are ordinary graviton polarization tensors. For further reading,
Chapter 7 of
Superstring Theory
by Green, Schwarz, and Witten [70] provides an especially enlightening discussion of the
four-point amplitudes in various string constructions.
As demonstrated by KLT, the property that closed string tree
amplitudes can be expressed in terms of products of open string
tree amplitudes is completely general for any string states and
for any number of external legs. In general, it holds also for
each of the huge number of possible string
compactifications [102,
103
,
49
,
50
,
86
,
3
].
An essential part of the factorization of the amplitudes is
that any closed-string state is a direct product of two
open-string states. This property directly follows from the
factorization of the closed-string vertex operators (5) into products of open-string vertex operators. In general for
every closed-string state there is a Fock space decomposition
In the low energy limit this implies that states in a gravity field theory obey a similar factorization,
For example, in four dimensions each of the two physical helicity states of the graviton are given by the direct product of two vector boson states of identical helicity. The cases where the vectors have opposite helicity correspond to the antisymmetric tensor and dilaton. Similarly, a spin 3/2 gravitino state, for example, is a direct product of a spin 1 vector and spin 1/2 fermion. Note that decompositions of this type are not especially profound for free field theory and amount to little more than decomposing higher spin states as direct products of lower spin ones. What is profound is that the factorization holds for the full non-linear theory of gravity.
![]() |
Perturbative Quantum Gravity and its Relation to Gauge
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |