82
3 Worldsheet Path Integral: Scattering Amplitudes
Expression with Ghosts
There are different ways to rewrite the 2-point amplitude in terms of ghosts. In all
cases, one correctly finds the 6 insertions necessary to get a non-vanishing result
since, by definition, it is always possible to rewrite the Faddeev–Popov determinant
in terms of ghosts. A first approach is to insert 1 =
d 2 z δ (2) (z) inside (3.32) to
mimic the presence of a third operator. This is equivalent to use the identity
0| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 |0 = 1
(3.45)
inside (3.33), leading to
A 0,2 (k, k
) =
C S 2
Vol K 0,2
V k (∞, ∞)c 0 ¯
c 0 V k (0, 0) S 2 ,
(3.46)
where V k (z, ¯
z) = c ¯
cV k (z, ¯
z). This shows that (3.16) can also be recovered using the
correct insertions of ghosts. The presence of c 0 ¯
c 0 can be expected from string field
theory since they appear in the kinetic term (10.115).
The disadvantage of this formula is to still contain the infinite volume of the
dilatation group. It is also possible to introduce ghosts for the more general gauge
fixing presented in [8]. An alternative approach has been proposed in [18].
3.2
BRST Quantization
The symmetries of a Lagrangian dictate the possible terms that can be considered.
This continues to hold at the quantum level, and the counter-terms introduced by
renormalization are constrained by the symmetries. However, if the path integral
is gauge fixed, the original symmetry is no more available for this purpose.
Fortunately, one can show that there is a global symmetry (with anti-commuting
parameters) remnant of the local symmetry: the BRST symmetry. It ensures
consistency of the quantum theory. It also provides a direct access to the physical
spectrum.
The goal of this section is to provide a general idea of the BRST quantization for
the worldsheet path integral. A more detailed CFT analysis and the consequence for
string theory are given in Chap. 8. The reader is assumed to have some familiarity
with the BRST quantization in field theory—a summary is given in Appendix C.2.
3.2.1 BRST Symmetry
The partition function (2.159) is not most suitable to display the BRST symmetry.
The first step is to restore the dependence in the original metric g ab by introducing
3 Worldsheet Path Integral: Scattering Amplitudes
Expression with Ghosts
There are different ways to rewrite the 2-point amplitude in terms of ghosts. In all
cases, one correctly finds the 6 insertions necessary to get a non-vanishing result
since, by definition, it is always possible to rewrite the Faddeev–Popov determinant
in terms of ghosts. A first approach is to insert 1 =
d 2 z δ (2) (z) inside (3.32) to
mimic the presence of a third operator. This is equivalent to use the identity
0| c −1 ¯
c −1 c 0 ¯
c 0 c 1 ¯
c 1 |0 = 1
(3.45)
inside (3.33), leading to
A 0,2 (k, k
) =
C S 2
Vol K 0,2
V k (∞, ∞)c 0 ¯
c 0 V k (0, 0) S 2 ,
(3.46)
where V k (z, ¯
z) = c ¯
cV k (z, ¯
z). This shows that (3.16) can also be recovered using the
correct insertions of ghosts. The presence of c 0 ¯
c 0 can be expected from string field
theory since they appear in the kinetic term (10.115).
The disadvantage of this formula is to still contain the infinite volume of the
dilatation group. It is also possible to introduce ghosts for the more general gauge
fixing presented in [8]. An alternative approach has been proposed in [18].
3.2
BRST Quantization
The symmetries of a Lagrangian dictate the possible terms that can be considered.
This continues to hold at the quantum level, and the counter-terms introduced by
renormalization are constrained by the symmetries. However, if the path integral
is gauge fixed, the original symmetry is no more available for this purpose.
Fortunately, one can show that there is a global symmetry (with anti-commuting
parameters) remnant of the local symmetry: the BRST symmetry. It ensures
consistency of the quantum theory. It also provides a direct access to the physical
spectrum.
The goal of this section is to provide a general idea of the BRST quantization for
the worldsheet path integral. A more detailed CFT analysis and the consequence for
string theory are given in Chap. 8. The reader is assumed to have some familiarity
with the BRST quantization in field theory—a summary is given in Appendix C.2.
3.2.1 BRST Symmetry
The partition function (2.159) is not most suitable to display the BRST symmetry.
The first step is to restore the dependence in the original metric g ab by introducing