170
GAUGE FIELDS AND STRINGS
Let US consider an immersion of the hypersurface described by the
functions:
(9.71)
The action must depend on
in such a way that it is invariant under
the transformation of diffeomorphisms:
or, explicitly:
=
a=U...,n
where the functions /" have to satisfy the condition:
= det
>0
(9.72)
(9.73)
This claim follows from the fact that x = x{^) and x = x(/(^)) represent
the same hypersurface, differently parametrized.
When we are looking for a continuous theory of surfaces, we have to
start from actions containing the minimal number of derivatives. If a
theory with such an action is renormalizable, then all the terms with
higher derivatives are irrelevant. Sometimes in order to achieve renormalizability one has to include higher terms. These questions we shall
discuss later.
A covariant expression with the minimal number of derivatives is just
the hypervolume A of our hypersurface. Therefore, the simplest action
which can be constructed is the following:
5[x(0] = m"oJd"«^(^))^/^
h(i) ^ detllMOII
M O = ^.x^ ^^x.
In principle, one can add some higher derivative terms like
d"i R(h)h^^^
(9.74)
Si = Cj
S2 = C2 d"(^ {A(h)xyh
1/2
(9.75)
etc.
(Here R(h) is the scalar curvature computed with the metric M and
A{h) is the corresponding Laplace operator). In each case special