String Theory and M-Theory

142 Strings with world-sheet supersymmetry
BRST symmetry
Superconformal field theory appears naturally when discussing the pathintegral
quantization of supersymmetric strings. In the quantum theory, it is
convenient to add Faddeev–Popov ghosts to represent the Jacobian factors in
the path integral associated with gauge fixing. Rather than discuss the pathintegral
quantization in detail, let us focus on the resulting superconformal
field theory.
As discussed in Chapter 3, in the bosonic string theory the Faddeev–
Popov ghosts consist of a pair of fermionic fields b and c with conformal
dimensions 2 and −1, respectively. These arose from gauge fixing the worldsheet
diffeomorphism symmetry. In the case of the RNS string there is also
a local supersymmetry on the world sheet that has been gauge-fixed, and
as a result an additional pair of Faddeev–Popov ghosts is required. They
are bosonic ghost fields, called β and γ, with conformal dimensions 3/2 and
−1/2, respectively. They have the OPE
Since these are bosonic fields, this is equivalent to
γ(z)β(w) ∼ 1
. (4.139)
z − w
β(z)γ(w) ∼ − 1
. (4.140)
z − w
The gauge-fixed quantum action includes all of these fields. It is S =
Smatter + Sghost, where Smatter is the expression in Eq. (4.130) and
Sghost = 1
2π
(b ¯ ∂c + ¯ b∂¯c + β ¯ ∂γ + ¯ β∂¯γ)d 2 z. (4.141)
The fields c and γ have ghost number +1, while the fields b and β have
ghost number −1. The bosonic ghosts β and γ are required to have the
same moding as the fermi field ψ µ – integer modes in the R sector and
half-integer modes in the NS sector. When the factors of z −h are taken into
account, this implies that ψ µ (z), β(z) and γ(z) involve integer powers of z
and are single-valued in the NS sector. whereas in the R sector they involve
half-integer powers and are double-valued.
The superconformal symmetry operators of this system are also given
as the sum of matter and ghost contributions. The ghost fields give the
following contributions:
T ghost
B
= −2b∂c + c∂b − 3 1
β∂γ − γ∂β, (4.142)
2 2
T ghost
F
= −2bγ + c∂β + 3
β∂c. (4.143)
2