284
GAUGE FIELDS AND STRINGS
(here VJl is the usual Chistoffel symbol,
normal vectors). Then the
intrinsic curvature R is connected with K by
R =
(10.84)
and is indeed a total divergence. However the separate terms in (10.84)
are not. Therefore we can write the following generalization of the
Nambu-Goto action:
5 = /^o
^0 J
(10.85)
It is easy to check that the second term is the only possible one (up to
total divergences) invariant under scale transformation jc -► Xx.
Adding this new term to the action is not a caprice. Its influence in
the infrared region determines the phase structure in the string theory.
So, if we want to compute the critical behaviour of random surfaces and
their geometrical and physical characteristics, it is absolutely necessary
to include this term in the action.
Our first goal will be to investigate the relevance of extrinsic
curvature in the continuum limit. Let us notice, that (10.85) can be
rewritten in other forms (modulo total divergences):
= Jg^' ^ (A(s)x)^
( 10.86)
Here:
- 1/2
Sag^'^g“ ’
à (g )x =
g
V ,n , =