2 3 6
GAUGE FIELDS AND STRINGS
Liouville action with fixed coupling (this is the only conformally
invariant expression of the required dimensionality).
We shall choose here a different approach to the problem, which
seems more meaningful. Let us generalize the expression for the ghost
lagrangian (9.352) which, without external field, has the form:
Sfh =
h__ d + co
= p
(9.374)
In this bosonic part, the fields co and h have the following transformation law:
w ^ - ( - l , 0 ) w " - ( 0 , -1 )
^(2,0) /!__ ^(0,2)
In general we could consider the case:
Sj. =
+b~
with
(9.375)
a ^ ^(j,0 )
a_
6^ -(1-7,0) b~
(0,7)
(0, 1 -7)
(9.376)
Our tactics now will be to couple to (9.375) an external gravitational
field not in the conformal gauge, in which the effective action is
nonpropagating, but in the gauge in which we have only one component of the gravition, say ^ . As we have already seen, this leads to a
finite expression containing g + ^ which can be generalized to a co variant expression.
First of all, we have to find an expression for the covariant derivative
in this gauge. The metric has the form:
ds^ = d r d r + ^ + ^ (d n "
= d r ( d r + ^ ^ ^ d n
(9.377)
It is easy now to find the coupling of the
-field to the a and ft-fields,
or, in other words, the energy-momentum tensor of the (a, b)-system.
The idea is to find the change of the action under the coordinate