172
GAUGE FIELDS AND STRINGS
As we have done before, let us begin with the computation of the
quantity
^ic(sX
=
(9.82)
which is equal to the number of immersions of a surface with fixed
intrinsic metric.
The use of the Lagrange multiplier representation gives:
A + iao
exp(
Jf[c, /i] =
X
d^x-d^x
(9.83)
The norms in functional spaces defining the volume elements in this
functional integral are given by;
WôxiOf = \h^^\ôx(OŸ d"i
X
d"(i
(9.84)
In deriving these formulas we have used, apart from locality and general
covariance, the claim that the distance in functional space has to be
invariant under
x(0 ^ x {0 + a(i)
A“'’(0 ^ A “*(0 + c‘'‘(0
(9.85)
where a and c “'’ are arbitrary. The invariance (9.85) of the measure
ensures the possibility of using the equations of motion for x and A,
which are derived just by the replacements (9.85) in the action.
It appears to be convenient to decompose
(9.86)
with
U i ) / “‘(0 = 0.