ATTEMPT AT A SYNTHESIS
259
These equations can be written in the form:
7 , v = 87re«^^
4> =
1^471
Stt
sr
J r
sr
s^
(10.28)
¿26^
A remarkable fact which we are now going to discuss is that this
effective action T is just the one which could be obtained by taking the
low energy limit of the string 5-matrix. Moreover, if we proceed to
higher terms in the jS-functions they would correspond to the higher
terms of the low energy expansion.
In order to clarify this amazing connection between the properties of
the world sheet of the string and the space-time into which it is
embedded, let us consider this problem in a slightly different context.
Let us suppose that we have a string, on the world sheet of which
“lives” some conformal field theory. Let us assume also, that in the set
of all operators of this conformal field theory there are operators {w„((^)}
with dimension two. If the resulting theory has zero total central charge
(after summing contributions from the coordinate of the string, the
ghosts and the extra conformal theory), the vertex operators of the form
(10.29)
describe scalar massless particles. We are going to demonstrate that the
effective action for these particles is related to the j8-functions.
Let us look at the perturbed lagrangian:
(10.30)
The operators u„ are supposed to satisfy the operator product expansion:
^
fnmiU,(0) + Icss singular terms
(10.31)
(where
are the symmetric structure constants of the operator
algebra which define the normalization of the three-point functions:
)
(10.32)