ATTEMPT AT A SYNTHESIS
2 6 3
In this integral we can at first consider all \^j\ R. Then we can apply
the operator product expansion (10.41) and get:
1
<«„(i?K.(iK,(0)...
d % ... dH,
d % ... d^^^<«„(R)u,(0)>.„ (10.44)
The finite part of the first factor in this formula is precisely equal to the
coefficient of the first power of the logarithm in (10.42), i.e. to the Pfunction. The second factor would be proportional to S„, if it had not
been for the contributions of ~ R in (10.44). These contributions lead
to
W
9m l() +
log— + ••
.„ = i ^ | !
Separating the finite part we obtain the desired relation:
dr((i>)
d(f>^ = Qn.imK4>)
(10.45)
(10.46)
from which it follows that the equations of motion are equivalent to the
condition of conformal invariance. We have not given a complete proof
of these formulas (for the reason that it has not been completed) but it
seems not too difficult to obtain this proof along the lines sketched
above.
The meaning of these results is quite transparent. They imply that if
the conformal field theory, which lives on the world sheet is unstable
under perturbations, then the string theory is unstable under condensation of the fields corresponding to these perturbations. If the twodimensional theory tends to a fixed point, defined by )?"() = 0, then the
original string theory will tend (after condensation of the fields) to a
new string theory, with a spectrum defined by the dimensionalities of
the operators at the fixed point. Stability in the sense of the renormalization group turns out to be stability in the sense of the mass spectrum.
In particular, the mass matrix given by the second derivatives of T,
according to (10.46) coincides with the matrix of anomalous dimensions, given by derivatives of jS, as it should be.
The only question which we have not settled yet is the central charge
of the resulting theory.
To investigate this problem let us introduce 5((^) the trace of the
energy-momentum tensor, and consider the pair correlation function in