160
GAUGE FIELDS AND STRINGS
finite dimensional case, if / is a group element then the Killing metric is
given by:
\\Sff = Tv{cD^l co = ¿ /./-^
This metric is invariant under the change:
(9.35)
(9.36)
(where dots mean the usual matrix multiplication.)
Diffeomorphisms, while forming an infinite dimensional group, behave very similar to the finite dimensional cases. The only crucial
difference is that in order to define the analogue of the trace in (9.35) we
have to use the metric of the manifold
as seen from (9.31).
Certainly, our manipulations with metrics in the functional spaces
make sense only if some invariant regularization and renormalization is
performed. This will be the case in all our future applications.
Returning to our problem we see that :
w \f)d f
(9.37)
C O = df
After rescaling t = T-f, a> ^ Te our expressions can be written as
2 _ (STf ■ +
i
(9.38)
W\\
I
dts\t)
From (9.38) we deduce that
Sihix) =
3>h(r)
m o
(9.39)