String Theory Demystified

CHAPTER 6 BRST Quantization 121
The conformal dimension [Eq. (6.4)] of the ghost fi elds follows from this
defi nition. Using the ghost energy-momentum tensor together with the energymomentum
tensor for the string we can arrive at the BRST current:
⎛ 1
jz () = cz () Tzz () z + Tgh() z
⎞
czT () zz () z c
⎝
⎜
⎠
⎟ = + () z ∂zc()()
z b z
2
The BRST charge is given by
Q
dz
i jz = ∫ ()
2π
Now, the central charge (i.e., the critical space-time dimension) comes from the
leading term in the OPE of the energy-momentum tensor, which is
D/
( z−w) The presence of this extra term is called the conformal anomaly since it prevents
the algebra from closing. So we would like to get rid of it. This is done by considering
a total energy-momentum tensor, which is the sum of the string energy-momentum
tensor and the ghost energy-momentum tensor, that is, T = Tzz () z + Tgh(). z It can be
shown that the OPE of the ghost energy-momentum tensor is
−13
Tgh() z Tgh( w)
=
( z−w) 2 4
Tgh( w)
wTgh( − −
( z−w) ∂ 2
w)
z−w 4 2
4
Taking the leading term in this expression to be of the form −( D/ 2)/(
z−w) , we
see that the ghost fi elds contribute a central charge of 26 which precisely cancels
the conformal anomaly that arises from the matter energy-momentum tensor. This
result actually follows from the nilpotency requirement (i.e., Q 2
= 0)
of the BRST
charge.
BRST Transformations
Next we look at BRST quantization by considering a set of BRST transformations
which are derived using a path integral approach. This is a bit beyond the level of
the discussion used in the book, so we simply state the results. Working in lightcone
coordinates, we defi ne a ghost fi eld c and an antighost fi eld b where c has