ATTEMPT AT A SYNTHESIS
2 6 1
We see from this formula that the jS-function can be expressed in terms
of the structure constants of the operator algebra. In higher orders in A
we would need structure constants of operator products involving
several w-operators.
Now let us return to the string theory. Scattering amplitudes in the
critical dimensionality are constructed out of simple vertex operators. If
a typical string resonance is described by the vertex operator
where p is the momentum, then on mass shell this operator has
dimension 2 and the scattering amplitude is given by:
(10.38)
This is almost the same expression as (10.34). The main difference is
that the short distance singularities governed by the operator product
expansion reveal themselves in the form of poles of
instead of
logarithmic divergences. The residues of these poles are precisely the
same structure constants which determine the jS-functions. Let us
consider the effective action for these massless particles, T((/>i,...,(/>„),
where {0^} are the fields corresponding to them. We presume that {0^}
are space-time independent, so that the particles carry zero momentum.
The expansion of T in (/> starts with the cubic term (since the quadratic
mass terms are equal to zero):
- n w ) ^ >.(/^xTW)Tr(i)n"(oo)> + •
(10.39)
We have recalled in the derivation of this formula what was always
implicitly assumed in the Koba-Nielsen integrals, that three integrations out of N are consumed by factoring out the 5L(2, C) group. So the
three point function does not have any integration at all.
We see from this formula that to this order the j?-function is
connected with the effective action :f
M ) =
dr
Wn
and, therefore, the condition for stability dV/d(l)„ = 0 coincides with the
condition of conformal invariance )?„((/>) = 0.
t Further conclusions arose in discussions with A. Zamolodchikov and partially use
his results.