194
GAUGE FIELDS AND STRINGS
We have:
(As)2,„ = e^<^>(A0L„ = ^
and on substituting this into (9.199) we get:
^ m ) = \og^- + (P(i)
(9.200)
(9.201)
This formula looks suspicious, since the left-hand side must be scalar
under conformal transformation:
^ ^(/(i)l/(0 )
At the same time:
cp(o-cp(m)) + \og d^
(9.202)
(9.203)
This is slightly confusing. We can even strengthen this confusion by
noticing that the Green function for non-coincident points,
^ (ÎJÎ2 )= - l0 g |^ l- Î 2
(9.204)
is not scalar either, since, for example under the scale transformation
^
it changes. The origin of the trouble is that the Laplace operator
on any compact manifold has a zero mode—the constant function. This
zero mode, which is revealed as translation invariance x -► jc -h a in the
integrand (9.192) must be carefully subtracted from the Green function.
The Green function without zero mode
is well defined and
does not increase logarithmically as (9.204). It is easy to show that
^'(ÎllÎ2) = ^(ÎllÎ2) + /(Îl) + /(Î2)
(9.205)
where f{^) is determined from the conditions that its Laplacian is
constant and that it cancels the logarithmic increase at infinity. It is easy
to give explicit formulas for /(¿), but they are not needed. Indeed:
i j
i j
+ HPiPum + fUj))
i.J
= iPi ■Pjmiij) + 2^p>md
= lP,-Pjm\^j)
(9.206)