84
3 Worldsheet Path Integral: Scattering Amplitudes
The second term in δ g ab is a Weyl transformation with parameter c w . Moreover,
b ab and B ab are not symmetric traceless.
The equation of motion for the auxiliary field is
B ab = i T ab := i
T
m
ab + T
gh
ab
,
(3.52)
where the RHS is the total energy–momentum tensor (matter plus ghosts). Integrating it out imposes the gauge condition g ab = ˆ
g ab and yields the modified BRST
transformations
δ = i L c
δ c
a
= i L c c
a ,
δ b ab = i T ab .
(3.53)
Without starting with the path integral (3.48) with auxiliary field, it would have been
difficult to guess the transformation of the b ghost. Since c a is a vector, one can also
write
δ c
a
= c
b ∂ b c
a .
(3.54)
Associated to this symmetry is the BRST current j a
B and the associated conserved
BRST charge Q B
Q B =
dσ j
0
B .
(3.55)
The charge is nilpotent
Q
2
B = 0,
(3.56)
and through the presence of the c ghost in the BRST transformation, the BRST
charge has ghost number one
N gh (Q B ) = 1.
(3.57)
Variations of the matter fields can be written as
δ = i [Q B , ,] ± .
(3.58)
Note that the energy–momentum tensor is BRST exact
T ab = [Q B , b ab ].
(3.59)