ATTEMPT AT A SYNTHESIS
257
It is straightforward now to compute the cut-oflF dependence of the
one loop partition function.
First of all, the logarithmic correction to the effective action arise
from the second term in (10.17):
Sili =
C d^k
ÌR M O ) daX^o
d^i J
(10.18)
As a result, we have renormalization of y^v(^) the form:
1
A
(10.19)
If the world sheet were flat, there would have been no other one loop
divergences. Counter-terms dependent on co„ do not arise, since they
can appear only through
4>lb =
- ^b^a +
^bif
which is a dimension 4 operator.
However, in the case of a curved world sheet there is another type of
counter-term which must be taken into account (as noticed by Fradkin
and Tseytlin). It is clear, that we have a dimension 2 object composed of
the external field
the curvature of the world sheet, R{^). So we can
expect counter terms, which do not depend on the derivatives of x, but
have the form:
{xmR{Og^'^ dH
( 10.20)
This kind of the logarithmic divergence already appears in flat space,
since the determinant of the Laplacian contains the term:
Tr log A ^ log A
( 10.21)
When we switch on
the coefficient in (10.21) gets modified. This
modification begins at the two loop order, for which we have to expand
the action (10.17) further. To give an idea of what happens, let us look
at the term (one of many) which appears in the expansion:
g^'^g“' ’K ^i,X xo (iW /
(10.22)