QUANTUM STRINGS AND RANDOM SURFACES
expression (after trivial change of scale):
175
K tiO '] = exp A
^x((^)exp( —
d,x-d^xd"n (9.966)
X (i(s)) = c(s)
The Green function for the contour c(s) is obtained by integration on
^Kt
^ab’
G[c(5)] =
exp - f i
X j*^ac(i) exp^ — J
d^x • d^^x
(9.97)
(where fi is the critical parameter).
Since expressions like (9.97) will form the basis of our further
discussion, it is worthwhile to present another derivation of it. Let us
consider the integral:
j m ,, txp(^-n j d " ^ - j
( 9 . 9 8 )
where QabiO is some tensor. The integral in (9.98) is supposed to be
covariantly regularized. That means that we can compute it by the
following procedure. First find the saddle point of the action (9.98):
4
d"i +
d"i^
+ 1 j h'i\h^“ g^,h‘ “ ’dK, + g,,0h^“ ) d"^ = 0
(9.99)
This equation gives the position of the saddle point and the value of the
action:
(9.100)
If we consider small fluctuations near this saddle point, we notice that
they are nonpropagating owing to the absence of derivatives in (9.98).