Next: Superstrings
Up: Perturbative String and Superstring
Previous: Spectrum of the Bosonic
Interacting Strings
So far we have described the free propagation of a closed (or open) bosonic string. In what
follows we consider the interaction of these strings. Here we focus in the closed string case,
the open case requires from further definitions. The interaction of strings at the perturbative
level is just the extension of the technique of Feynman diagrams for point particles to extended
objets. The vacuumvacuum amplitude is given by

(10) 
The interacting case requires to sum over all loop diagrams. In the closed string case it means
that we have to sum over all compact orientable surfaces. In two
dimensions these surfaces are
completely characterized by their number of holes (the genus). The
relevant topological invariant is the Euler number
, where is the scalar curvature of the worldsheet . In order to
include the interaction of strings, the generalization of the Polyakov action consistent with
its symmetries is given by

(11) 
where is an scalar background field and represents the
gravitational coupling constant of the twodimensional EinsteinHilbert Lagrangian.
in the above equation stands for the geodesic curvature of .
If we define the string coupling constant by
then Eq. (10)
generalizes to

(12) 
The amplitude defined onshell correspond to and the rest () corresponds to
loop corrections.
The definition of correlation functions of operators requires of the idea of vertex
operators
. These operators are defined as

(13) 
where
(with being a generic massless field of the bosonic
spectra) is a local operator assigned
to some specific state of the
spectrum of the theory. For instance for the tachyon it is given by
. While that for the graviton with polarization
it is given by
Local operators
are diffeomorphism and conformal invariant and therefore more convenient
to define scattering amplitudes.
Thus one can define the scattering amplitude of the vertex field operators by
their
corresponding invariant operators
. In perturbation theory the
scattering amplitude is given by

(14) 
This scattering amplitude is, of course, proportional to the correlation function of the
product of invariant operators
as follows

(15) 
truecm
Effective String Actions
In order to make contact with the spacetime physics we now describe how the
spacetime equations of motion come from conformal invariance conditions for the
nonlinear sigma model in curved spaces. The immediate generalization of the Polyakov
action is

(16) 
where is an arbitrary background metric of the curved target space . The
perturbation of this metric
in the partition function
, leads to
an expansion in powers of This partition
function can be easily interpreted as that coming from the interaction of the
string with a coherent state of gravitons with invariant operator
with
.
On the other hand, the Polyakov action can be generalized to be consistent with
all symmetries and with the massless spectrum of the
bosonic closed strings in the form of a nonlinear sigma model

(17) 
where is the target space curved metric, is an
antisymmetric field, also called the KalbRamond field, and is the scalar
field called the dilaton field. From the viewpoint of the twodimensional
nonlinear sigma model these background fields can be regarded as coupling
constants and the renormalization group techniques become applied. The computation of the
quantum conformal anomaly by using the dimensional regularization method, leads to
express the energymomentum trace as a linear combination

(18) 
where are the oneloop beta functions associated with each coupling
constant or background field. They are explicitly computed and give

(19) 

(20) 

(21) 
where
Weyl invariance at the quantum level implies the vanishing of the conformal anomaly
and therefore the vanishing of each beta function. This leads to three coupled
field equations for the background fields. These conditions for these fields can been
regarded as equations of motion derivable from the spacetime action in dimensions

(22) 
where is a normalization constant.
It is interesting to see that a redefinition of background metric under the
transformation in dimensions
with
leads to the background action in the
`Einstein frame'

(23) 
where
and
The form of this action will of extreme importance later when we describe the strong/weak
coupling duality in effective supergravity actions of the different superstring theories
types. In the above action
is the
gravitational coupling constant in dimensions, i.e.
A very close procedure can be performed for the open string and compute its effective low
energy action. It was done about 15 years ago and for gauge fields with constant
field strength it is given by the DiracBornInfeld action

(24) 
Later we shall say something about some applications of this effective action.
truecm
Next: Superstrings
Up: Perturbative String and Superstring
Previous: Spectrum of the Bosonic
root
20010115