234
GAUGE FIELDS AND STRINGS
/i+_ = 0 and we have two components
and /i__ only, while the
condition (t% = 0 implies also that we have only two components / +
and f-oi the vector-spinor /®, which correspond to spin 3/2, while the
spin 1/2 component is removed. We must remember that all our
computations of covariant derivatives have been done on the background where we have some metric p but no gravitino field In the
final results we shall be able to restore /-dependence by the use of
supersymmetry. It should be noted, however, that it is easy to work
with / present. All we have to do is to consider a superspace generalization of (9.361) and (9.362).
To put things together, we see that the ghost contribution to the
functional integral is defined by the determinants, which arise from the
following eigenvalue problems (recall (9.352)):
pd^ip ^w+) = i£/i+ +
= g^~
= p~^ d
+ = ìEco^
for bosons, and
= p-^
ìEe^
(9.364)
(9.365)
for fermions. If we apply
to the first equations of each pair we get:
(9.366)
— p ^ d_{p
u) = E^u
—
d_{p^'^ d^v) = E^v
^ 0.-l) = P
t^(0,-l/2) = P
The general operator acting on tensors with conformal spin (0, j) would
be
(9.367)
For 7 = 0 this is the scalar laplacian: for j = —1/2 and — 1 these are
ghost operators, which we have just now derived. We shall need one
more value, j = 1/2, corresponding to Dirac fermions. Indeed, Dirac
fermions transform as:
tA ± -'(d r)
±\-l/2 , ( 1/ 2, 0)
(0, 1/ 2)
and hence the Dirac eigenvalue problem is:
p
(9.368)
(9.369)