QUANTUM STRINGS AND RANDOM SURFACES
191
Hence:
- i ( 5 + /i++)co_5_w.) - i/i++(5+ (9.184)
Since only the < propagator is nonzero, the third and the fourth
terms do not contribute.
We are left with
der*'^(L • « = J ^ 0)^ exp-/ —
ld + a>+d_(o.
-ih^^(d _ co ^)d ^o )_ ^}
(9.185)
From this it is easy to obtain the quadratic term of the effective action
for /i+ + :
s„ = - ‘
+ Aq)h^ -4) n«.‘_.. _(9);
k + q
k + q
k + q
n»_‘_,--(9)= O +a_ O + 5- O a_
k
k
k
d^k / 1 /cl(/c -h 1
kl.{k + q).
-^2 — q.
( 2 n y \\6 k \ k + qy
1
k_(k + q)_
-7 7 ^ -
16 ^ k \ k -f qy
16
k^(k + qy
A computation similar to (9.160)-(9.163) gives the result
Yigh_
16^^. I 24tc 4n 4n I 4Snl6q^
(9.186)
(9.187)
Substituting into (9.186) and comparing with (9.165) we conclude that
the role of ^ is now played by the number 26 and in the gauge (9.180)
det^^^ (L^L) = e x p |—
Jd^z (2d^(p d^q> -h ¡1^ e’’) |
(9.188)
Collecting now (9.176), (9.177), (9.180) and (9.188) we get the final
expression for the distribution of the surfaces in internal metrics in
(9.156):
7closed surf. _ r
r 26 - ^ r
)
^(P expj -
J
(i (d,