String Theory and M-Theory

SOLUTION
3.2 BRST quantization 75
After two successive conformal transformations w(u(z)), one finds
(∂w) 2 T (w) = T (z) − c c
S(u, z) −
12 12 (∂u)2S(w, u),
where ∂ = ∂/∂z. In order to prove the group property, we need to verify
that
This can be shown by substituting
S(w, z) = S(u, z) + (∂u) 2 S(w, u).
dw
du =
du
dz
−1 dw
dz
= w′
u ′
and the corresponding expressions for the higher-order derivatives
d2w du2 = w′′ u ′ − w ′ u ′′
(u ′ ) 3
d3w du3 = w′′′ (u ′ ) 2 − 3w ′′ u ′′ u ′ − w ′ u ′′′ u ′ + 3w ′ (u ′′ ) 2
(u ′ ) 5
into S(w, u). ✷
3.2 BRST quantization
An interesting type of conformal field theory appears in the BRST analysis
of the path integral.
In the Faddeev–Popov analysis of the path integral the choice of conformal
gauge results in a Jacobian factor that can be represented by the
introduction of a pair of fermionic ghost fields, called b and c, with conformal
dimensions 2 and −1, respectively. 8 For these choices the b ghost
transforms the same way as the energy–momentum tensor, and the c ghost
transforms the same way as the gauge parameter.
These ghosts are a special case of the following set-up. A pair of holomorphic
ghost fields b(z) and c(z), with conformal dimensions λ and 1 − λ,
respectively, have an OPE
c(z)b(w) = 1
ε
+ . . . and b(z)c(w) = + . . . , (3.76)
z − w z − w
while c(z)c(w) and b(z)b(w) are nonsingular. The choice ε = +1 is made
8 For details about the Faddeev–Popov gauge-fixing procedure we refer the reader to volume 1
of GSW or Polchinski.