QUANTUM STRINGS AND RANDOM SURFACES
231
The meaning of the last expression in the brackets is that it is
“traceless” in the sense that being multiplied by d" it gives zero. All this
is quite analogous to the bosonic case (9.113). Accessibility of the gauge
depends on the possibility of solving the equation
(LpS) = ^„8 -
= (P a
(9 .3 4 7 )
where cp^ is an arbitrary spin 3/2 spinor with (r^cp^ = 0. This is to be
combined and compared with the bosonic part of this equation (9.113):
(Lgco) =
(9 .3 4 8 )
with arbitrary traceless h^b. As we shall show now, the ghost determinants are just those of the operators
and Lp. In the purely bosonic
case we have already derived this result (remember that we found a
det(L^ Lp) factor in the functional integral). It is almost obvious that
for the fermionic case we have to add dQt~^(Lp Lp). Here we shall
derive this result by a different (from the bosonic case) method, which
seems quite useful to know.
Let us take a super-metric (g, Xa) the superconformal gauge
9ab — P^ah
l a , a = i ^ a l ) a
(a is a spinor index) and consider the Faddeev-Popov equality:
(9 .3 4 9 )
^co^eSig - L^oji)6{Xa - Lps)
(9 .3 5 0 )
which serves as a definition of W. Inserting this into the functional
integral for the partition function we get
(9 .3 5 1 )
where we have omitted integration over the supercovariance group
Q)ioQ)8. Now, it is convenient to introduce ghost fields explicitly by
representing:
e"'^(^’^“) = det L„ d ef^ L .
exp{-{K L g c o ) - ( / , L p s ) } (9 .3 5 2 )
The signs ( —) and (+ ) here indicate that we integrate over fields with
reversed statistics: co and h are fermions, while e and / are bosons.